@Johan, I agree with you: Differential Geometry of all kinds is locally Euclidean. It seems nothing has changed some thousands years in this knowledge field.

@Artur, actually I have worked in the 'soliton concept' many-many years from now. It seemed to me very promising then, but I didn't follow that step. I have also worked with chaos theory, but I have to remind you: in order to find the equilibrium points it is necessary to linearize our dynamical system first!

@Marek, as linear I mean the procedure we use: we are always take the Calculus path, which is always locally a linear mapping. I agree, everything that cannot represented with a basis is difficult! We need more tools...

@Alessandro, you are the exception of the rule, since you are working with nonlinear RQA!

@M. Thomas, you have mentioned a very common and useful linearization process.

@Hemantha, a good example is the Proof of Riemann integrability theorem for 'exotic' partitions of xi's.

@Afaq, when we face what you have mentioned we are all happy!

@Ulrich, probably this is the reason why the 3-body problem is unsolved: it is not linear!

No, in simple linear equation we have chaos (for any parameters). In the results we have new methods for analyzing complex systems (technical, economic,…), for example Sharkovskii's theorem for nested orders, Mandelbrot’s fractal systems and another.

Soliton theory is one example of substantially nonlinear science, see e.g. https://fas.org/sgp/othergov/doe/lanl/pubs/00326980.pdf and http://en.wikipedia.org/wiki/Soliton for a first introduction to the subject.

Multisoliton solutions which are exact solutions of certain (solitonic) PDEs are quintessentially nonlinear objects. By the way, some solitonic PDEs (e.g. the Ernst equation) appear as reductions of the Einstein equations of General Relativity and soliton techniques made it possible to produce new exact solutions to these equations and hence of the (fully nonlinear rather than just linearized) Einstein equations. The soliton theory methods also enabled one to find the celebrated multi-instanton solutions of the Yang--Mills equations for the gauge fields.

On the opposite end of the spectrum, chaos theory also deals with essentially nonlinear systems, see e.g. http://en.wikipedia.org/wiki/Chaos_theory and http://fractalfoundation.org/resources/what-is-chaos-theory/ for starters, although,, as correctly pointed out in the other answer, the chaos may occur in the linear systems as well.

Demetris, you are right: we are linearizing almost everything. But why? Are we just lazy or we don't want to see non-linearities? No, the real reason is that we haven't invented yet proper tools to deal with non-linearities, that's why. While we know precisely how many solutions there are for a given set of linear equations (algebraic or differential), nothing similar can be said when it comes to the systems of non-linear equations.

But we are aware of non-linearities and we are trying hard to deal with them. The Navier-Stokes equation is more than 100 years old. We cannot pretend there is no hysteresis in various phenomena - obvious non-linearity.

I wouldn't say that our science is linear. Look at any the so called scientific calculator, at its "function keys". I see only non-linear functions there.

On the other hand, approximate ("linearized") solutions are of quite practical value, and are within reach.

In my opinion the word 'Linear Science' is highly ambiguous, for example if I use Recurrence Quantification Analysis (RQA) this is surely a non-linear tool because it is based on distance functions that, being based on circles (the circonference is the locus at a given distance from a centre), and I can do a lot of interesting ting with this method, nevertheless if instead of X,Y coordinates I put the system in a a coordinate system having as axes the distances from given poles all appear (and can be intended) as linear, the same (reversed) applies to principal component analysis that is a linear technique but that can be very good to study non-linear systems: simply the non-linearity appears as distributed over different component axes.

All in all I do not think it is useful to speak of linear and non-linear science: I could go to a linear representation after a non-linear transformation to make the system easier to appreciate...attached an example in which a non-linear problem (the derivation of mutual position of European cities starting from distances from some reference poles) is resolved by a linear application (PCA) over a non-linear (distances) space, is it Linear or non-Linear Science ?

I have sympathy with the question : Linear and non Linear are relative concepts . Most often non linear systems can be transformed in to linear systems for simplicity . Taking a simple example, an exponential function can be transformed into logs and linearized in logs . To explain in a philosophical sense -of course ,one has to look at many dimensions of an issue and in that sense non linear understanding is needed: for example , when I say this is Day time , it is Night time somewhere-else in the world!

As far as Statistics is concerned, all data other than Circular and Spherical data, are linear. Accordingly, Euclidean spaces are enough to deal with the statistical matters at least.

Dear Demetris Christopoulos, Non-linearity is reality. And, many techniques can handle those non-linear systems. But an interesting point is that the resolution of digitization almost diminishes the concept of non-linearity. Hence fully satisfied with our linear science.

no, I'm not satisfied with this paradigm. And I say in my articles, urgent need to change the paradigm.

Dear Afaq,

The question was on whether we should be satisfied regarding remaining within the Euclidean dimensions. We count linearly; we do not count non-linearly. Should we possibly start counting non-linearly? This is the point Demetris has raised if I have understood the question correctly..

Dear Professor Hemanta Baruah, Thanks for your point. I am with engineering background and answered the question with my knowledge in the field. Most of the time we experience non-linear signals in the forms of transients, errors, noise, .. and we deal those non-linearity and accommodate in our A to D conversions. I think about your point 'Euclidean dimensions' and can say strongly "satisfied" because in Digital Communication, Wireless Communication, Antenna and Wave propagation multidimensional situations occur.

Dear Professor Ahmad,

Some data, for example directional data, are non-linear. But basically, statistical analysis is for linear data. That is why I answered in that way. In your field too, the data which you collect from laboratory experimentation are linear. They are multidimensional, but still 'linear'.

Dear Professor Hemanta Baruah, In some situations our data are non-linear. We model those and handle using state variable approach.

Scientist should not indulge in nonsensical buswords like 'centuries of Linear Science'.

Where is Linear Science in the 3-body problem?

Do not all these ways of local linearization you mention show that primarily non-linear structures are under consideration.

@Johan, I agree with you: Differential Geometry of all kinds is locally Euclidean. It seems nothing has changed some thousands years in this knowledge field.

@Artur, actually I have worked in the 'soliton concept' many-many years from now. It seemed to me very promising then, but I didn't follow that step. I have also worked with chaos theory, but I have to remind you: in order to find the equilibrium points it is necessary to linearize our dynamical system first!

@Marek, as linear I mean the procedure we use: we are always take the Calculus path, which is always locally a linear mapping. I agree, everything that cannot represented with a basis is difficult! We need more tools...

@Alessandro, you are the exception of the rule, since you are working with nonlinear RQA!

@M. Thomas, you have mentioned a very common and useful linearization process.

@Hemantha, a good example is the Proof of Riemann integrability theorem for 'exotic' partitions of xi's.

@Afaq, when we face what you have mentioned we are all happy!

@Ulrich, probably this is the reason why the 3-body problem is unsolved: it is not linear!

Dear Demetris,

Thank you. Meanwhile, the three-body problem is perhaps going to open a Pandora's box now!

Dear Hemanta, in case that somebody could attempt to touch the 3-body problem, here are some links:

http://www.scholarpedia.org/article/Three_body_problem

http://www.mat.uniroma3.it/users/chierchia/FILES_DIDATTICA/AM8/delaunay.pdf

Especially see page 10 of:

http://www.math.utexas.edu/users/jjames/hw4Notes.pdf

Obviously is a NONLINEAR problem, but it can be 'linearized' around the libration points, see page 15 and 21 of the same link.

I think this emphasis on 'linear' against 'non-linear' issue is particualrly vivid in continuous methods, if you work on discrete (like nayone working in data analysis by statistical techniques such as PCA or RQA) the point is much less crucial, you never rely heavily on some functional form,the rquirement of 'stationarity' is for exaple a much demanding task that makes you to decide for one method or the other...in the attached paper this is particularly clear.

Alessandreo, at Fig.13, page 260 of your rqatedesco.pdf, it is shown that RQ plot can finfd the maximum of a function. Have you tried to find the inflection point? Is there any relevant article? Thanks.

Dear Demetris,

The Pandora's Box has remained closed! Well!

A lot of themecially in applicative areas like the detection of changes in dynamical regime of chemical reactors:

http://www.sciencedirect.com/science/article/pii/S0098135403002059

http://www.sciencedirect.com/science/article/pii/S037596010200436X

or in financial time series

http://www.sciencedirect.com/science/article/pii/S0378437106010685

epilepsy

http://www.sciencedirect.com/science/article/pii/S0375960104014501

mathematical non-linear models:

http://www.sciencedirect.com/science/article/pii/S0375960196007414

ecology

http://www.sciencedirect.com/science/article/pii/S0304380006006223

and many others.....

But the question was on non-linear data, and not on non-linear models.

Dear Hemanta, I think Pandora' s Box has a small period of disappearance!

Some thoughts:

1)Any differentation is a linear mapping, so whenever you are using Calculus or Differential Geometry (or GR) you are quietly doing linear science, although your model (ODE, PDE, ...) seems to be nonlinear.

2)Since our local universe is locally Euclidean, then every linearized version of the true explanation can also describe it very well

3)In QM we are using the linear superposition principle: |Psi>=Sum(a_n * |psi_n>,n=1..infinity), so it is again linear science

So, do you really believe that we have develped a genuine non linear science?

Http://www.cs.nyu.edu/~roweis/lle/papers/lleintro.pdf

Dear Demetris Christopoulos, What is your opinion about Boolean Difference? because you are talking about ODE, PDE but not mentioning about BD.

Dear Afaq, please read this:

http://www.enel.ucalgary.ca/People/yanush/publications/frei96.pdf

Dear Demetris Christopoulos, Thank you very much for this article.

I have been quite fascinated by this topic. I once asked the question (in my PhD thesis) about the price we pay in performance when we choose to look for linear solutions. To my amazement, it appears that many practitioners in the controls area, willingly or unwillingly exclude most of the solution space with this approach and at times may be far off from finding the optimal solutions as a result. From an educational perspective because linear systems are easier to teach and grasp, most of our engineers graduate knowing little else and later find themselves enslaved by these limiting linear concepts.

Dear Kelly, if you think deeper about the fact that everybody talks about 'quantum this and quantum that' every time he/she wants to impress people and based on the fact that QM is 100% linear science, then what exactly are we doing except from applications of Linear Algebra?

Demetris,

considering quantum theory 100% linear science is a widespread misconception. In quantum field theory in wich dynamics is defined by non-linear equations for operator fields we have the two aspects: operators act as linear operators in a Hilbert space but their dynamics is determined by a non-linear equation between the operators. In non-fieldtheoretic formulations the non-linearity is not so manifest but is there, as can be seen from the fact that many-body quantum theory is ('by second quantization') equivalent to a field theory. So quantum theory combines linear and non-linear mathematics in an ingenious manner.

@Ulrich, the 100% is about the linear operators involved, the tools, of course not that everything in QFT is linear (even the Coulomb potential is not linear). Since I am interested for the tools, I think that QFT is a linear science.

Demetris,

are you aware that Hamiltonian classical mechanics can be formulated in terms of linear operators in real Hilbertspace (Koopman)? This incorporates even the classical three body problem into linear science!

Actually, this example shows that it is not unimportant which structure in a theory is linear. In classical mechanics the harmonic oscillator 'is linear' since its equation of motion is linear. The anharmonic oscillator is non-linear! In this spirit, free quantum field theory is linear and interacting QFT is non-linear. The sad fact that no such theory was so far defined in a mathematically exact way may well be attributed to this non-linear nature.

Johan,

are there neverheless days when you consider it an ingenious edifice? Then we agree on the matter.

Johan,

its a possibility for you too!

Johan,

did you ever hear about an experiment that shows that the partial derivatives of electric and magnetic fields in vacuo are related as stated by Maxwell's equations?

What convinces physicists of the truth of these equations is:

1. They gave correct results for directly measurable quantities of interest (such as antenna radiation patterns) in all tested cases.

2. They arise from a convincing theoretical analysis. Convincing not necessarily for the public, but for those who understand also the potential alternatives.

Exactly these criterea let the electro-weak part of the standard model (to which the Higgs field/particle belongs) look trustworhy to most of the informed physicists.

@Ulrich, you are openning another important topic: that of the experimental testing of our Holy Readings in Physics. Does the non existence of a negative experiment prove that a theory is experimentally well tested? Or, just that the hidden aspect still remains hidden?

Demetris,

the alternative is not but

'that there might still be a hidden aspect waiting for discovery'. And, whoever will get hold of such an aspect and will turn it into a physics experiment will become a famous physicist. Isn't this a fair game?

@Johan

We sit on too distant trees for this discussion to lead to a useful conclusion.

Johan,

is the anomalous magnetic moment of the electon absurd physics?

@Johan,

there are many trained theoretical physicists who write papers which rightfully no peer-reviewed journal is willing to publish. Who sees subtraction of infinity from infinity as the bottom line of renormalisation does probably not write good papers.

Johan,

I simply tried to express that the case you brought up is compatible with my experience without assuming unjustified censorship.

That renormalisation is not mathematically defensible is a defensible statement. However, also Dirac's 'delta function' was for some years not mathematically defensible, till 'distributions' were introduced by Mathematicians. Physics always had a wider view of acceptable formalisms than mathematics.

You can formulate renormalisation in a way that all quantities under consideration are well-defined and finite and after introduction of well-motivated 'counter-terms' remain finite in the limit of removed cutoffs. Such counterterms can by no means found for every conceivable field theory. Fermi's field theory of weak interactions was of this unfavorable type ('non reormalisable'). The electroweak part of the standard model turned out to be renormalisable (work of 't Hooft and Veltman, Nobel prize 1999). This looks not like the work of morons to me (to come back to one of your earlier qualifications).

Johan,

I tried to tell you that renormalistion is unable to 'to get the results that you want'. You only can get results that in an, admittedly cryptic way, are preformed in the non renormalized form of the theory.

But I see that your worldview is immutable and I will not further waste my time trying to change it.