Consensus in Science and Sports: An Inconvenient Truth

The idea of scientific consensus is used by the alarmists of global warming and by the wikipedians controling the information on Wikipedia. However, scientific truth cannot be determined by majority voting, only by scientific facts and arguments. One fact or argument by one scientist can outweigh the consensus of billions of people. Scientific consensus can lead astray, since it can give the false impression of scientific truth, when it is only the superstition of many.

Political truth in democratic societies is determined by consensus of the majority,  but scientific truth should not be determined by consensus, in particular not by third-party majority consensus, but by real combat between active living scientists. 

It is the same in sports: The Wimbledon matches between Borg and McEnroe were not determined by consensus of the spectators, but by Borg and McEnroe alone. 

Borg and McEnroe represent the active living scientists carrying the scientific knowledge at any given time, who by playing matches of science or disputations in seminars and journals decide the current truths. 

In sports, you win by walkover if your opponent does not show up to the match, and it is the same in science. You cannot defend your position by saying nothing, neither can the songbird  defend its territory by singing nothing.

An illustration is given by the match about how to resolve  d'Alembert's paradox in fluid mechanics which has been going on for 255 years. This long match has now come to an end by the publication of my resolution together with Johan Hoffman in the leading Journal of Mathematical Fluid Mechanics. The victory is declared by Google putting our resolution in top position in a search on "resolution of d'Alembert's paradox". It is a walkover victory because the entire editorial board of the Journal of Fluid Mechanics says nothing.

d'Alembert's paradox of zero drag in inviscid flow is important since much of modern fluid mechanics is related to the paradox in one way or the other, as explained in my knols on fluid mechanics.

Michel Crichton expresses the essence very clearly: 

• Let's be clear: the work of science has nothing whatever to do with consensus. Consensus is the business of politics.  Science, on the contrary, requires only one investigator who happens to be right, which means that he or she has results that are verifiable by reference to the real world.  In science, consensus is irrelevant. What is relevant is reproducible results. The greatest scientists in history are great precisely because they broke with the consensus.
• The work of science has nothing whatsoever to do with consensus. There is no such thing as consensus science. If it is consensus, it isn't science. If it's science, it is not consensus. Period."

Strings and String Wars

Not Even Wrong Peter Voit gives his view of the current status of string theory in a conversation about Philosophy and String Wars with philosopher Craig Callender:  

• the basic equations of string theory are not known
• and besides are so incredibly complicated that nobody can understand them even if they were known 
• no predictions come out from the unknown equations
• progression has stopped
• physics departments have stopped hiring string theorists
• last hope is that LHC will give new input but LHC does not work...

As concerns the role of mathematics in string theory Voit informs us: 

• the mathematics is so sophisticated that only Witten can understand
• but the problem is not so much sophisticated mathematics that nobody understands
• it is that the physical idea of vibrating strings in 10 dimensions does not seem to work out
• physics departments now are saturated with sophisticated math and look for physics instead and string theory moves to math departments. 

There is lot in this discussion connecting to my previous posts on physics such as Illusions of Theories of Everything and Is Crazy-Physics = Pseudo-Science?

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.