About chaos we could said from perspective of epistemic or ontic state. For epistemic subject, chaos will be something non cognitans, when any conection is not posiible to recoognize. From ontic perspective we could said as state of world, but I do not know what it could be, chaos for whom ? If epistemic subject does not exist then chaos is not recognize, but now I could said from ontic pperspective that every particle get in to any other particles, then chaos = everythings possibilities. I do not agree that everything is possible. Becaouse:

Our mind, learning about reality, makes its description with the help of the intellect. Twardowski drew attention to the fact that there exist objects such as sensory objects in the mind space, it can be considered a perception of hallucination, which I find rather next to the main problem raised by me in this text. I am going to talk about the physical basis of empty empty terms, or rather why they can be considered empty concepts. This is related to the phenomenon of superposition, in which the particles have their values ​​bounded, i.e. they have, for example, color, hardness, are positive, negative, give mass. Thus the set of X particles, type z, with properties P, Q, R (z), will interact with the set of particles Y, type u, with the properties G, H, J (u) in the manner of W. For this reason, as Kant wrote, "Not everything that is possible, is real." We can make any kind of alphabet in the word formation that will have no reference, which designate will not exist, while Hume wrote about the ability to fantasize, as a capacity for any combining sensory impressions in our mind, as opposed to the ability to imagine, as a reproductive act of recalling objects that our mind has seen before. To sum up: the fact that particles have properties and interact with each other at the microscopic level and do not combine freely, but due to properties, result in certain aggregates at the mesoscopic level and the ability to fantasize, are the empirical basis for the existence of empty concepts within the linguistic space.

Thank you, Piotr, for your rather long feedback! Which book of Twardowski should I read? And that of Kant (about that 'possibility')? Marc.

I guess your question is more philosophy than mathematics but, anyway, I'll give my opinion from a scientific point of view.

Nonlinearity means a nonlinear response of a system to a certain input. Most of the systems (e.g. stars) can be described as nonlinear systems since a linear relation is too simplistic to describe complexity in nature. Chaos emerges as a consequence of the nonlinearity of systems, meaning that subtle changes on the parameters describing the system might lead to a radically different states after some time. However, for short intervals of time nonlinear systems behave linearly. In fact, linearity can be considered (or have to be IMHO) as a special case of nonlinearity where the onset of chaos occurs at infinite time.

There is also a mathematical result, the shadow lemma, which guarantees that, in spite of chaos, the description of a nonlinear system is a computable problem.

I found it in psychology textbook.

I 'll answer the two added answers sequentially:

[1] Javier Pascual-Granado, Dear Javier,

(a) as far as possible I am doing my utmost best as a philosopher of science: both philosophy and science is relevant for me.

(b) chaos is fractalizable, i.e. geometrically to be smalled (like a particle e.g.), but it remains geometry, and geometry is utterly discrete, and discreteness amounts to linearity; that is exactly my question.

[2] Piotr Grabowski, Dear Peter,

(a) a single question: why do you find my question one from the psychology textbook? I expect your more extensive answer. Thanks in advance! Marc

The problem concerning nonlinearity vs linearity is really subtle one. Besides philosophical deliberations we have concrete results referring

to this issue in the theory of dynamical systems. Namely, it can be shown that finite dimensional nonlinear dynamical systems can be brought down

to infinite-dimensional linear ones. Such approach is known as the Carleman linearization (see the monograph: K. Kowalski and W.-H. Steeb, Nonlinear Dynamical Systems and Carleman Linearization, World Scientific, Singapore, 1991). For an easy illustration of the Carleman linearization (or

Carleman embedding) consider the simplest nonlinear one-dimensional ordinary differential equation

dx/dt = x^2

On defining the sequence

x_n := x^n

we arrive at the linear infinite-dimensional system of the form

dx_n/dt = nx_{n+1}

Clearly x=x_1, so the nonlinear equation is reduced to the infinite dimensional linear system, more precisely, it is embedded in the latter one.

The Carleman technique was generalized to the case of partial differential equations and difference equations. For the furter development

of the Carleman approach I recommend my monograph: K. Kowalski, Methods of Hilbert Spaces in theTheory of Nonlinear Dynamical Systems, World Scientific, Singapore, 1994). I finally point out that in the case of the Hamiltonian system, the counterpart of the Carleman linearization is the Koopman approach applied in the ergodic theory.

Krzysztof Kowalski, it doesn't seem very useful to have an infinite dimensional system in practice. Do you know any practical application of Carleman linearization?

An example of practical applications are nonlinear areonautics problems discussed in: B.W. Gaude, Solving nonlinear aeronautical problems using the Carleman linearization method, Sand report, SAND20001-3064, Sandia National Laboratories. In this report one can read: ,,Fortunately, Kowalski and Steeb summarized a large portion of this work into one book ... The Carleman embedding technique is the theoretical method used to approximate solutions of the nonlinear aeronautical problems studied in this research. Kowalski and Steeb's book was used extensively to derive

the linearization.''

'linearization' is one of the tools to study nonlinearity. In the process, some information of the system is lost but at least some information about very difficult to understand nonlinear systems, is indeed obtained using this approach.

The problem sounds as follows: does a nonlinear a priori system under regard allow a suitable transformation resulting in the system linearization? The answer in general is No... So, the speculations about the linearization of every nonlinear system are not more than nonapty jookes!

Dear all,

(i) I got a little twig from the literature you gave me. Thanks! It is really subtle, but not a speculation nor a fiction, but it regards imagination that is necessary for any theory-formation!

(ii) As philosophers of science (supposedly all of you) we do not leave such theoretical problems only to philosophers to grapple with but should try to solve such problems. This is one of them. (And it could finish with infinitization every practical minded scientist, you and me, hates).

(iii) 'linearization' is one of the tools to study non-linearity. That is right (according to me), Govind! (see the description of my question); and it has implication to more problems, a.o. the limitedness itself!

(iv) what is a nonlinear a priori system, Anatolij? Please give me a more extensive answer. Thanks! Marc.

Thank you Andreas! I 'll look at the Carleman technique: isn't the same as the Carleman linearization? Thanks for your answer! Marc.

Our minds may like to think linearly and that's why it is difficult to predict things beyond the next few moments. Some may recall the IBM manager who (maybe by 1950) thought that there would be no market for Computers, because the 5 Computers already existent then were already able to solve all problems of the world. OK, his prediction was “a little more” than a bit wrong.

I happen to deal with real-world nonlinear systems and I can tell you that their behavior is not linear! If you want to know the behavior of a nonlinear system (described by a nonlinear differential equation) in the close neighborhood of its equilibrium point, then, yes, linearization might help. However, to know whether the nonlinear system will reach the equilibrium point at all or not, you better learn some nonlinear systems theory, stability theory, etc.

Dear Itzhak,

Thank you for your answer. Please give me a human behaviour or experience which is non-linear! Thank you. Marc.

Dear Marc,

Sorry, but, although I happened to play Prof and Dr., I am first of all an engineer and my main job has always been to put things to work, and also all the time better and faster. Therefore, because I thought that your question was talking about non-linearity versus linearity in general, I was taking about nonlinearity in machines and other systems.

Take the robot for example. The motion of Its arms is ruled by sinusoidal functions. Because, for small angles, sin(x) is close to x, then for a first understanding, one can use a linear model. However, for real control, one must use the more realistic (and more complex) non-linear model.

Dear Itzhak,

Thank you for the concise but clear answer. Your work as an engineer is praiseworthy. However, it is what mathematicians are doing: the instrumental reasoning. And instrumental reasoning can only be executed if it is limited because distinction presupposes limitedness. Well, linearity and non-linearity presuppose distinction (see e.g. fractals). The conclusion is up to you.

Thanks anyway for the answer! Marc.

Dear Marc,

I am afraid my answer was not as clear as you write.

Your "this is what mathematicians are doing" by which you seem to distinguish between mathematicians and engineers "who simply work" may show some misunderstanding.

If you mean purely-purely theoretical things, with their result being a paper, you could be right.

In order to make sure that your robot does the precise work that it is supposed to do, or that the plane that you fly brings you safely to wherever you need, I can assure you that engineers need, learn, teach, and may even have to develop tools in pretty high levels of Math, in particular nonlinear differential equations.

Dear Itzhak,

Thank you for the sincere answer. Let me answer concisely your added answer as follows; if it is not true, please get back to me:

(i) both mathematicians and engineers use Math (see your last paragraph);

(ii) that "simply work" regards both (see e.g. Nagel & Newman on Goedel's Proof);

(iii) your purely-purely theoretical things is not very clear to me; please explain it to me. Thank you in advance. Marc.

Dear Marc,

I had and have no intention to enter an argument, although, in relation to other questions, it did happen (even without intention :-).

By purely theoretical, I meant new mathematical ideas, such as a new topology, etc., that mathematicians may develop and where I would not interfere.

Practice forces me to continuously learn, understand and use, and even to further correct or develop delicate points of mathematical methods. Same practice made me aware of nonlinearities where linear approximations simply do not work.

Simple free fall distance is d=g*t2. After the first msecs, maybe linear approximation does not give you a bad result, yet no much afterwards. Also, if a stone is dropped, you'd better feel it immediately after it fell then after a good few seconds. :-)

All the Best,

Itzhak

Dear Itzhak,

Thank you for the short but clear answer. My question can simply be shortened as follows: Is science (I mean: mathematics and physics) exclusively based on limitedness upon which distinction can be made?

Thanks again for your added reply! Marc.

Dear Marc,

I happen to think that no one (at least, not me) has the general solution for the general problem. You asked a very specific question on linearity versus nonlinearity, which happened to touch on things which have managed to preoccupy me,

To me, the new question looks more like the philosophy of science. I am not even sure I really understand it, so I’d better rest my case here and leave it to others, who might have interest and might have dedicated some thought about it.

Have Fun!

Itzhak

Mathematics is limited by the axioms we choose, the rules we use to derive statements from those axioms, and our own ingenuity. Everything else is is either conjecture or reality.

Personally, mathematics is much more *real* to me than reality, but YMMV.

Read Goëdel's theorem, then ponder your questitin again ... I think you'll come to the conclusion that it is undecidable.

Dear Itzhak,

Thank you again for your sincere answer! Marc.

Dear Randall,

Please explain to me further what you mean by .

Thank you for the answer. Marc.