Role of Mathematics Education in Society??

Can we learn something about the role of mathematics education in our society, from how mathematics departments present their educational programs?  The following statements are typical: 

• Princeton University: The mathematician's best work is ART, a high perfect art, as daring as the most secret dreams of imagination. Mathematical genius and artistic genius touch one another. (Gösta Mittag-Leffler).
  
• MIT: Mathematics provides a language and tools for understanding the physical world around us and the abstract world within us. 
• University of Chicago: One of the wonderful things about the University of Chicago is that EVERYONE has to take mathematics. Most students complete this requirement by taking one of our calculus sequences. 
• Chalmers University of Technology: The mathematical sciences are fundamental and indispensable to a large part of modern science and engineering. Progress in other disciplines is often linked to an increased use of mathematics. Mathematics is however also a subject in itself, and fundamental research is a necessary condition for its many applications.
• University of Washington: Mathematics is both a science and an art. Like any great art, mathematics has an intrinsic beauty and coherence that has attracted practitioners for centuries. Yet, unlike other arts, mathematics is a surprisingly effective tool for describing the natural world. Indeed, mathematics has come to serve as the foundation of modern science, through its language and results. Some mathematical results were initially developed in order to solve internally generated mathematical problems and only later found application in other disciplines; other mathematical results were inspired by the needs of these other disciplines. The two facets of mathematics - tool of science and subject of inquiry for its own sake - have come to be interwoven into a complex fabric. 
• University of Oxford: Mathematics plays a pivotal role in the progress of society and its continued growth relies on the exchange and development of research ideas, the encouragement and teaching of the next generation of mathematical thinkers, and outreach to the public and schools. 

We  summarize:

• Mathematics is a form of sublime art, which miraculously has shown to be very useful in science and engineering. EVERYBODY needs to learn mathematics in order to understand the physical world outside and abstract world inside ourselves.

We observe that the existence of the computer is not visible. 

The message is that mathematics primarily is a form of art which by developing according to its own inner principles, to which computing does not belong, best serves the needs of society. In this scenario there is little incentive to reform motivated by the computer now changing the society outside mathematics departments and their educational programs. 

Obama on Unicorn in FEniCS

Unicorn is a unified continuum mechanics fluid-solids interaction solver in FEniCS. The web is now being flooded with images and paintings of Obama riding a Unicorn, which signifies that Obama has joined the FEniCS development team. Wellcome!

Obama's mathematics for a health care reform is now debated: Will it add to the budget deficit? Republicans accuse Obama of breaking his deficit-neutrality promise and health experts scoff at the Democrats' fuzzy math.

In The Obama Education Plan: Math, Science and Rigor Obama's spokesman  says:

• We have 20 English teachers for every job and can't find a math teacher, and that's a problem. We have to pay math and science teachers differently.
• We've got to push to get more kids taking algebra in eighth grade, and then you start to think about calculus in that fourth year of high school.
• One downside of No Child Left Behind was 50 states dumbing down things.
  

In his first major speech on education on March 10 this year, Obama states:

• It's time to give all Americans a complete and competitive education from the cradle up through a career. We've accepted failure for far too long. Enough is enough. America's entire education system must once more be the envy of the world -- and that's exactly what we intend to do.
• Studies show that children in early childhood education programs are more likely to score higher in reading and math, more likely to graduate from high school and attend college, more likely to hold a job, and more likely to earn more in that job. 
• See, I want every child in this country to have the same chance that my mother gave me, that my teachers gave me, that my college professors gave me, that America gave me.

Math is supposed to bring the US back into its proper position in the World, but which math will do it? Traditional Calculus?

The Tower of Mathematics Education

In a critical analysis of school mathematics, Sverker Lundin identifies the gap between what school mathematics claims to offer the students, and what is actually offered, a gap which I will argue originates in university mathematics education. A gap which today can be bridged by the computer.

Primary school mathematics is a bleak simplified version of secondary school mathematics, which is a bleak simplified version of college and university mathematics. The ideology of mathematics education is formed on the highest level, which in bleak simplified versions is propagated down to secondary and primary levels. 

The ideology of university mathematics as presented by mathematics departments offering mathematics education, is based on the following postulates:

• mathematics is the language and basic tool of science
  
• mathematics is a prime expression of human intelligence and creativity 
• the World is governed by mathematical laws

rooted in Greek mathematics and boosted in the scientific revolution initiated by 

• Newton's law of gravitation
• formula for solution of differential equation of two-body problem,

expressing the power of 

• Calculus: derivatives, integrals and differential equations
• linear algebra: arithmetization of geometry

which has formed the basis of university mathematics for more 100 years. 

The goal of mathematics education is to show that mathematics is a universal tool with universal applicability. In Calculus/linear algebra this is demonstrated by

• generality of the concepts of limit, continuity and derivative
• list of primitive functions of elementary functions
• formula for solutions of simple differential equations
• generality of linear algebra,

aimed at expressing that all problems can be described by algebraic/differential equations which can be solved analytically in terms of elementary functions. A gap results from the fact that in reality few differential equations can be solved analytically. 

In primary education the universality can be expressed in

• regula de tri: proportionality
• Euclidean geometry
• formula for solution of 2nd order algebraic equation

where regula de tri represents a universal method for solving a wealth of problems of proportionality. A gap results form the fact that few problems are simple problems of proportionality or can be formulated as a second order algebraic equation.

The gap between fiction and reality of mathematics education comes from the fact that analytical mathematics is not universal: Already the three-body problem lacks an analytical solution formula. This gap undermines the crediblity of mathematics education with the effect that most students fail to get the main message that mathematics is universal with a potential to solve a universe of problems.

However, computational Calculus/linear algebra obtained by adjoining the computational power of  computer, has universal capabilities, which opens new possibilities of closing the gap by bringing ideology and practice together in a modern meaningful mathematics education ...the goal of the Bod&Soul project...see knols on mathematics/science education...

New Blog on School Mathematics

Language and mathematics are commonly viewed to be the core subjects of school education following all students during 10-12 years of training. It is obvious that good language skills are helpful in coping with life and job, but most people use so little mathematics that 10 years of study seems out of proportion. How can it be that so much time and effort is spent to so little effect?

In his Ph D thesis Sverker Lundin uses tools of sociology and psychoanalysis in a critical analysis of the nature and role of school mathematics and now continues the discussion on a blog. As of now the language is Swedish but let's hope that an English version will soon be available. The analysis and discussion is very important,  very intriguing and even mind-boggling! Check out yourself!

If You Are Looking for Trouble

If you are looking for trouble, opportunities seem to be offered in the big-physics experimental projects of the Large Hadron Collider LHC and the Laser Interferometer Gravitational-Wave Observatory LIGO. LHC is constructed to detect the Higgs boson and particles and forces that reigned earlier than the first trillionth of a second of time. LIGO is constructed to detect gravitational waves. 

LHC does not work, while LIGO works but does not detect anything but noise. New York Times reports in Giant Particle Collider Struggles:

• The biggest, most expensive physics machine in the world is riddled with thousands of bad electrical connections.
• After 15 years and $9 billion, and ashowy “switch-on” ceremony last September, LHC has to yet collide any particles at all.
• But scientists say it could be years, if ever, before the collider runs at full strength, stretching out the time it should take to achieve the collider’s main goals, like producing a particle known as the Higgs boson thought to be responsible for imbuing other elementary particles with mass, or identifying the dark matter that astronomers say makes up 25 percent of the cosmos.
• The collider was built to accelerate protons to energies of seven trillion electron volts and smash them together in search of particles and forces that reigned earlier than the first trillionth of a second of time.
• But some physicists admit to being impatient. “I’ve waited 15 years,” said Nima Arkani-Hamed, a leading particle theorist at the Institute for Advanced Study in Princeton. “I want it to get up running. We can’t tolerate another disaster. It has to run smoothly from now.”
• Pauline Gagnon, an Indiana University physicist who works at CERN: “The public pays for this, and we need to start delivering.”

Earlier than the first trillionth of a second after Big Bang! Can you come up something more mind-boggling?

In Latest News form the LIGO Laboratory we read:

• This fall, the ground-breaking fifth science run of the LIGO interferometers came to a successful conclusion as an integrated full year's worth of science data was accumulated with all three LIGO interferometers in coincidence. The interferometers--antennae tuned to detect the delicate ripples of gravitational-waves coursing through the fabric of space-time--have achieved unprecedented sensitivity and astrophysical reach, as well as excellent duty cycle.
  
• LIGO now moves into its next phase of progress, Enhanced LIGO, an ambitious project to improve LIGO’s sensitivity by a factor of 10 and increase its source reach (the number of extragalactic sources within its range) by a factor of 1000. Advanced LIGO is expected to observe gravitational-wave signals from such cosmic sources as merging pairs of black holes and neutron stars at the rate of several each month. Because the characteristics of the source of the gravitational waves is encoded in the waves themselves, Advanced LIGO will pioneer a new form of astronomy utilizing gravitational waves to observe and interpret some of the most violent events happening in the universe. Construction on this stage will begin early in 2008, with first observations using the Advanced LIGO configuration scheduled to commence in 2014.

LIGO did not detect any waves. What are the prospects that Enhanced LIGO will do it? If gravitational waves do exist, why are they so terribly difficult to detect? 

The cost of the big-physics projects LHC and LIGO is big, more than $15 billion together, but so far the result is zero. How much money will have to be put in to get anything but zero out? 

How much is $15 billion? The cost of the US health care reform is estimated to $1 trillion, which is also about the total cost of US wars since 2001. The cost of LHC and LIGO thus so far is less than 2% of the war cost. In either case the result seems close to zero...and nobody knows if spending $1 trillion on big-physics will suffice...

Reductio Ad Absurdum

In the interview LHC and Unified Field Theory in response to the question: Is it possible as you drill farther and farther down with bigger and bigger accellerators and collisions, you find tiny particles made by tinier particles, or do you see a limit to this reductionism? Frank Wilczek, Nobel Prize in Physics 2004, states:

• It's an enchanted enchanting world...to me that makes it more bearable...
  
• Protons and neutrons were studied very carefully, but we learned that protons are very complicated objects that obey very complicated equations...you don't get beautiful simple equations like Newton's law or Coulombs law...these marvellous simple equations, when you work at the level of protons and neutrons... 
• But we discovered that protons are made quarks and gluons which obey simple equations,  for which I got the Nobel Prize: Maxwells equation on steroids, that's what it is... You can write them down in a few lines. That's big progress.
• I think we are going to hit bottom. The equations we have are very rigid: you cannot change them without making them worse. We get this wonderful unification. We are really on the right track.
• One can guess that Nature likes beautiful equations...has the same perception of beauty as we have as trained physicists...

This is extreme reductionism or reductio ad absurdum. 

If we inside a Russian doll find a simple smallest microscopic doll without any features at all, does it carry the secrets of the bigger macroscopic dolls with many features? Recall that Newton's and Coulomb's marvellous simple laws are macroscopic laws far from reductio ad absurdum... 

Scientific Consensus = ??

Science is the religion of our time, and like all religions it can be used in a more or less constructive way. The role of scientists is to use the scientific method to gain knowledge that at best is useful for people on our planet. 

In the debate on global warming, Al Gore and many with him, argue that there is a scientific consensus on the causes of global warming.  But what do we mean by scientific consensus? Wikipedia, the incarnation of consensus, states:

• Scientific consensus is the collective judgement, position, and opinion of the community of scientists in a particular field of study. Consensus implies general agreement, though not necessarily unanimity. Scientific consensus is not by itself a scientific argument, and it is not part of the scientific method. Nevertheless, consensus may be based on both scientific arguments and the scientific method.
• In public policy debates, the assertion that there exists a consensus of scientists in a particular field is often used as an argument for the validity of a theory and as support for a course of action.
  

This captures pretty well a consensus about scientific consensus: It is not by itself a

scientific argument or part of the scientific method, but can nevertheless be used to support action.

A poll showing that a 75% of general audience believes the human activity can influence the global climate, would then give support to scientific consensus on the causes of global warming. 

But since scientific consensus may be more politics than science, it can lead to wrong actions. So how is then scientific consensus about global warming formed? In the Wikipedia article Scientific opinion on climate change we read:

• National and international science academies and professional societies have assessed the current scientific opinion on global warming...attributable to human activities...endorsed by the Intergovernmental Panel on Climate Change (IPCC).

The many international science academies and professional societies backing the consensus, all base their opinions on IPCC.  Now IPCC itself motivates its assessment from a couple computer simulations showing a better fit with certain data with a certain forcing, than without forcing.

In previous blogs I have tried myself to assess the accuracy of these computer simulations, and found more questions than answers. The mere fact that forcing changes the output is not surprising, since this is built into the code. The mere fact that you get a headache by bumping your head into a wall, does not prove that if you have a headache, then it is caused by bumping your head into a wall. It may be caused by drinking too much wine or...

We thus find scientific consensus on global warming ultimately resting on one single set of computer simulations. But a top-down pyramid is very unstable and a different set of computer simulations will motivate a different scientific consensus. The consensus itself has no scientific value.  It is like political opinion, which can shift from one day to the next. In fact, there is no scientific consensus on causes of global warming not even today, as shown in Climate Debate Daily.

Again we come to the conclusion that what is needed is more reliable computer simulations, not more scientific consensus based on unreliable simulations.

Another example of scientific consensus is given in my blogs on d'Alembert's paradox

and Interview with Editors of Journal of Fluid Mechanics showing the fluid mechanics community scientific consensus expressed by Wikipedia and the Editor of JFM: First, there is no paradox. Second, if after all maybe there is a small paradox, it can somehow be resolved by hinting at the viscosity of a viscous fluid. 

The scientific consensus is thus that a basic problem of fluid mechanics, is not a basic problem of fluid mechanics, but this is in contradiction to the scientific method and thus corrupts the science of fluid mechanics. If the science of global climate is equally corrupt, the we may be in trouble...for more trouble see Giant Particle Collider Struggles.

In the motivation for the 2007 Nobel Peace Price awarded to Al Gore and IPCC we read:

• Through the scientific reports it has issued over the past two decades, the IPCC has created an ever-broader informed consensus about the connection between human activities and global warming. Thousands of scientists and officials from over one hundred countries have collaborated to achieve greater certainty as to the scale of the warming. Whereas in the 1980s global warming seemed to be merely an interesting hypothesis, the 1990s produced firmer evidence in its support. In the last few years, the connections have become even clearer and the consequences still more apparent.

Ever-broader consensus, thousands of officials collaborated, firmer evidence, greater certainty, even clearer, still more apparent...

The Threat of Computed Climate

Suppose a team of mathematicians today announces that the Earth will be hit by a large meteor in 5 years and 20 days, according to their computations. How should we react?Ask about the reliability of their computations, I guess. If we then get the answer that the computations are so complicated that there is no way we can check that they are correct. How are we then to react? Should be panick and launch a massive exodus to Mars? Well, this is all fiction and has not yet happened, so we don't have to worry.

Suppose a couple of teams of mathematicians today are telling us that according to their computations, in a few decades the sea level will rise by 5 meters and 20 centimeters and billions of people will have to move to avoid getting drowned, in a great exodus. Again we ask about how the computations are performed and what their reliability may be? But again we get the answer that the computations are so complicated that nobody outside the teams of mathematicians involved, can understand them. How are we then to react?

And this is not fiction: Global warming from burning of fossil fuels is today predicted by a couple of computer codes simulating the global climate. One of these codes is the National Center for Atmospheric Research Community Atmosphere Model CAM 3.0 also referred to as CCSM  presented as follows:

• CCSM belongs to an elite category of computer-based simulation models known as general-circulation models. Such models use mathematical formulas to recreate the chemical and physical processes that drive Earth's climate. What emerges from trillions of computer calculations is a picture of the world's climate in all its complexity. 

The future of mankind seems to be governed by this code. What is then the reliability/accuracy of CCSM? Is there anyway we can tell or is CCSM so complex that its beyond scrutiny by people outside the coding team?

Well, the first thing is to take a look at the documentation to learn that the numerics involves 

the following fixes of an otherwise unstable or overly stable useless code: 

• time filter
  
• horisontal diffusion correction
• initial divergence damping
• mass fixers
• energy fixer
• monotonicity fixer.

Further, the codes uses mathematical formulas to model the following physics:

• deep convection
• moist convection
• precipitation
• cloud fraction
• short wave radiation
• long wave radiation
• surface exchange
• vertical diffusion
• boundary layers
• sulfur chemistry
• prognostic of greenhouse gases

plus a number of other fixes and models. The most advanced version CCSM3 uses a horisontal computational grid of mesh size of about 150 km x 150 km and about 30 vertical levels, which must be regarded as a coarse grid requiring a lot of difficult mathematical modeling. The numerics seems to be a concoction of rather old-fashioned techniques.

Looking at this information, as an expert of computational fluid dynamics, my scientific self protests. It does not seem right to base far-reaching limitations on energy consumption on such a code. Or is the accuracy of the code irrelevant, as long as it serves to limit the use of energy?

What if a code tomorrow shows that CO2 can increase without negative climate effects? What if this code is wrong, but we don't understand that?

Of course we should be careful how we use resources on our only Earth, but believing that we will be hit by a meteor in 5 years and 20 days may not be the best basis for rational decisions, if the meteor is only fiction predicted by an incorrect computer simulation.

The numerics could be upgraded using modern adaptive finite element methods with flexible meshing and a posteriori error control. This may not do wonders, but the effort can not be spared...For the first time in history the future of mankind is in the hands of computational mathematicians... 

Sheep Herd Accuracy?

The next UN Climate Conference will take place in Copenhagen in December under Swedish chairmanship of EU. UN Climate chief Yvo de Boer hopes the conference will in particular reach agreements to limit the growth of emissions in developing countries, required to be necessary by predictions of catastrophical global warming.

The most advanced computational methods for long-time predictions of the global climate, the AOGCM coupled Atmospheric Ocean General Circulation Models, use computational grids of size 

• surface: 2.5 x 3.75 degrees, height: 19 levels HadAM3
  
• surface: 1.4 x 1.4 degrees, height: 26-40 levels CCSM3

to simulate turbulent convective radiative reactive phenomena. This means a surface grid of 30 x 30  to 60 x 60 for a quarter hemisphere including complex geometry of land and ocean.

From my experience from computing turbulent flow like the flow around a car or wing, the resolution of HadAM3 is definitely not impressive, and the grid of CCSM3 also seems too coarse. For such grids turbulence modeling is required, which however is an open problem.

What about the accuracy of these models? According to IPCC TAR underlying the Kyoto Protocol

• The model mean exhibits good agreement with observations.
• The individual models often exhibit worse agreement with observations.
  

The logic here is to take the mean of many incorrect models, to get a correct model! As a scientific principle this is wonderful, because you can generate so much knowledge out of ignorance: 

• The mean value of many incorrect values, is the true value!
  

But of course you say: What about the sheep herd effect? Just because all sheep rush in certain direction, does it mean it is the right direction? Not even in politics, and definitely not in science.

Mathematics of Global Warming?

One of the few dissidents in the climate debate is Lars Bern coauthor with Maggie Thauersköld of the Chill-Out - The Truth about the Climate Bubble, which questions the accepted truth that global warming is such a threat to humanity that in particular poor people should be prevented from increasing their energy consumption.  

Decisions aimed at controling global warming are based on predictions obtained by computational solution of basic equations of physics expressing conservation of mass, momentum and energy. Correct decisions require correct predictions and the question of the accuracy and reliability of the predictions must be addressed.

The Intergovernmental Panel of Climate Change IPCC  states in its Fourth Assessment Report (AR4) released in 2007 about Climate Models and their Evaluation:

• One source of confidence in models comes from the fact that model fundamentals are based on established physical laws, such as conservation of mass, energy and momentum,  along with a wealth of observations.  
• A second source of confidence comes from the ability of  models to simulate important aspects of the current climate. 
• A third source of confidence comes from the ability of models to reproduce features of past climates and climate changes. 
• Nevertheless, models still show significant errors. 
• This is partly due to limitations in computing power, but also  results from limitations in scientific understanding or in the  availability of detailed observations of some physical processes.  Significant uncertainties, in particular, are associated with the  representation of clouds, and in the resulting cloud responses  to climate change. Consequently, models continue to display a  substantial range of global temperature change in response to  specified greenhouse gas forcingof model development, they have consistently provided a robust  and unambiguous picture of significant climate warming in response to increasing greenhouse gases.
• In summary, confidence in models comes from their physical basis, and their skill in representing observed climate and past  climate changes. Models have proven to be extremely important  tools for simulating and understanding climate, and there is  considerable confidence that they are able to provide credible quantitative estimates of future climate change, particularly at larger scales. 
• Models continue to have significant limitations,  such as in their representation of clouds, which lead to uncertainties in the magnitude and timing, as well as regional details,  of predicted climate change. 
• What does the accuracy of a climate model's simulation of past or contemporary climate say about the accuracy of its projections of climate change? This question is just beginning to be addressed...
  
• Nevertheless, over several decades  of model development, they have consistently provided a robust  and unambiguous picture of significant climate warming in response to increasing greenhouse gases.
• Over several decades of development, models have consistently provided a robust and unambiguous picture of significant climate warming in response to increasing greenhouse gases.

We read that IPCC states that

• the models still show significant errors
• models continue to have significant limitations,

which expresses that the accuracy of the predictions is very questionable, according to the highest authority of IPCC. Compare with a Global Climate Modeling: Fate of Humanity?

The statement that the confidence comes from the fact that the models have physical basis, shows an alarming innocence as concerns computational mathematical modeling. Physical basis? What else? Religion or parapsychology?

This statement is aimed at impressing the uninformed, but is a triviality to the informed.  Is the message that the poor should feel the pressure of established physical laws and not ask for more?

The statement cited last is the key: models show response to increasing greenhouse gases.

Yes, but this is because they are so designed. Models are not reality. 

It may be that there is antropogenic warming derived from human activities, but statements that this is proved by simulations using Global Climate Models can be questioned.

 

If mathematical predictions are to serve as the basis of decisions influencing the lives of billions of people, the accuracy has to be improved. Suddenly nothing seems more important to humanity than computational mathematics, fluid dynamics and simulation technology ...because this is the only way to get information about the future we know of ...

Compare Climate Debate Daily and Resisting Climate Hysteria, Global Warming - Sensibilities and Science and Global Warming - A Peculiar Issue by Richard Lindzen stating:

• What was done by IPCC, was to take a large number of models that could not reasonably simulate known patterns of natural behavior (such as ENSO, the Pacific Decadal Oscillation, the Atlantic Multidecadal Oscillation), claim that such models nonetheless accurately depicted natural internal climate variability, and use the fact that these models could not replicate the warming episode from the mid seventies through the mid nineties, to argue that forcing was necessary and that the forcing must have been due to man. 
• The argument makes arguments in support of intelligent design sound rigorous by comparison.  It constitutes a rejection of scientific logic, while widely put forward as being ‘demanded’ by science.
  

If we reject scientifc logic, what shall we then rely on?