[Short summary: You use the scale that maeks more sense and is more easily applied. Just stay aware of the limitations of your modell and you will be fine. The main problem you seem to hint at is the inability of the human mind to process complex systems most of the time.]

Dear Yuri,

this is not so much a question of what is possible, but rather one of what is necessary. Although that is not the whole answer. When modelling ecosystems, you usually come accross the question of scales (and weather to use discrete or continuous ones) from two different perspectives.

1. What do I measure? What am I able to measure? What do I want to measure? Continuous data requires very precise measurement. But to get an idea of what is going on, even to prove some mechanics/theorems, you do not need continuous data. Why would you want to be more complicated than you need to?  A good example is the application of "Kraft's Classes" in determining the relative competitive advantages in tree communities. Competition among trees is most often a function of height, because taller trees use up Photosynthetic Active Radiation (PAR) first. You could measure every tree's height and its position relative to each other. But it is not necessary. Instead you simply put them into one of 5 classes and have an almost perfect prediction of their development in the next decade.

2. Does it make a difference when I want accurate information? Using discrete steps is much easier and sometimes even mandatory (think of modelling male and female animals, continuous doesn't make any sense here). Think of modelling hunting and you Modell consecutive shots as vectors. Maybe the exact mean will be above the target, but that does not mean that you scored any hit.

Modelling is the art of reducing complexity as much as possible while still getting meaningful results. In ecology as well as social siences you usually do not have accurate enough data on the complex system that is reality. You do not gain much by using continuous data. Even if you get a more exact Modell, you won't get any more information from it.

There is a good chapter on this issue in Grimm and Railsback 2006.

While modellers (even in some in economics, I hope) are aware of the limits of their models, these or other heuristic versions of them might lead to false conclusions in the event of facing a complex problem. Complex problems were defined by Dietrich Dörner along five lines: The number of elements is high. The elements are interconnected. The System changes over time. The elements show adaptive behaviour and have diverging/different goals. The amount of information is too big.

Reducing this complexety leads to undercomplex models like Equilibrium theory in economics. To avoid getting trapped in "false" models, you need to be open to apply different models to the same Problem. And to make the circle complete: Some of them might be discrete and others continuous. It is enough to know the discrete stops of a bus line. In case the bus breaks down, you have to Switch to a continuous model. There are no "true" models. And there is no need to reduce the world to discrete or continuous models.

Best regards

https://books.google.de/books?id=12MvUbMeog8C&redir_esc=y

[Short summary: You use the scale that maeks more sense and is more easily applied. Just stay aware of the limitations of your modell and you will be fine. The main problem you seem to hint at is the inability of the human mind to process complex systems most of the time.]

Dear Yuri,

this is not so much a question of what is possible, but rather one of what is necessary. Although that is not the whole answer. When modelling ecosystems, you usually come accross the question of scales (and weather to use discrete or continuous ones) from two different perspectives.

1. What do I measure? What am I able to measure? What do I want to measure? Continuous data requires very precise measurement. But to get an idea of what is going on, even to prove some mechanics/theorems, you do not need continuous data. Why would you want to be more complicated than you need to?  A good example is the application of "Kraft's Classes" in determining the relative competitive advantages in tree communities. Competition among trees is most often a function of height, because taller trees use up Photosynthetic Active Radiation (PAR) first. You could measure every tree's height and its position relative to each other. But it is not necessary. Instead you simply put them into one of 5 classes and have an almost perfect prediction of their development in the next decade.

2. Does it make a difference when I want accurate information? Using discrete steps is much easier and sometimes even mandatory (think of modelling male and female animals, continuous doesn't make any sense here). Think of modelling hunting and you Modell consecutive shots as vectors. Maybe the exact mean will be above the target, but that does not mean that you scored any hit.

Modelling is the art of reducing complexity as much as possible while still getting meaningful results. In ecology as well as social siences you usually do not have accurate enough data on the complex system that is reality. You do not gain much by using continuous data. Even if you get a more exact Modell, you won't get any more information from it.

There is a good chapter on this issue in Grimm and Railsback 2006.

While modellers (even in some in economics, I hope) are aware of the limits of their models, these or other heuristic versions of them might lead to false conclusions in the event of facing a complex problem. Complex problems were defined by Dietrich Dörner along five lines: The number of elements is high. The elements are interconnected. The System changes over time. The elements show adaptive behaviour and have diverging/different goals. The amount of information is too big.

Reducing this complexety leads to undercomplex models like Equilibrium theory in economics. To avoid getting trapped in "false" models, you need to be open to apply different models to the same Problem. And to make the circle complete: Some of them might be discrete and others continuous. It is enough to know the discrete stops of a bus line. In case the bus breaks down, you have to Switch to a continuous model. There are no "true" models. And there is no need to reduce the world to discrete or continuous models.

Best regards

https://books.google.de/books?id=12MvUbMeog8C&redir_esc=y

Dear Klemens, I see your point. I fully agree that we should reduce complexity and the quantity of measurements. But I am talking about thinking. When we measure  sine wave, it is enough to get 4 points per period to reconstruct it. But you have a basis of functions in continuous space to start a Fourier transform!

Let us talk about ecology. A mathematician with continuous thinking will write a kind of logistic growth model for the dynamics of population in a given ecological niche. Or write a dynamical system of Lotka-Volterra equations for modelling the dynamics of predators and prey. Then make few measurements to find best fit for parameters. In reality, population is always discrete, but if it is large enough, discreteness does not matter asymptotically.

A computer modeller without any idea of continuity will invent a complex interaction model with 100 wolves and 1000 rabbits and will die in the details of random matches, that will require more and more computing time without a possibility to get any significant conclusion. I know that some economists write such models with 1000 (or more) of interacting consumers and then require more and more powerful computers for its processing. Understanding asymptotics and limit theory along with the basics of economics (equilibrium, game theory, etc) allows to save both time and brains.

There is a joke among physicists: "A theorist will easily solve a problem about stability of a table with one leg, then with infinite number of legs - and spend the rest of his life trying to solve that stability problem for a table with 4 legs!" Now the situation has changed. Computer allows to solve the problem with 4 legs in finite time, but people are often unable to solve the problem with infinite number of legs and to prove that the case of 1000 of legs (and 1 million) is very similar, because this should be done without computer.

But I am more concerned that there are models (see Krugman) with only 2 points in space and infinite variety of commodities in each. A very wrong abstraction of reality, to my mind. And how can this be named "new economic geography"? Real geographical space is a continuous 2-dimensional manifold. Its discretization depends on the observed pattern (here spatial Fourier transform is useful) but cannot be done a priori.

Dear George, I see that you are both mathematician and statistician by education. You are right that empirical observations are often treated using statistical methods. Especially in medicine, where it is hard to track determinism due to complexity and non-observability. Maybe the God knows if this patient died because of self-emerging critical blood tension at a certain point in time or because of surgeon's error in a certain action. Statistics will tell us that the risk to die for this operation is, let say, 10%.

In  physics randomness often plays a minor role and observed patterns are 99% deterministic. Then statistical methods of data treatment are less important. However, we have statistical physics to prove that certain random processes can be very well described by deterministic equations.

Dear Yuri,

I think you recognize that the problem you raise is more fundamental than discrete and continuous models. You are assuming that the nature of reality is one or the other, which is an unknown even in the physical sciences.

But by thinking, I assume you mean what practictioners do with the sample of information they have at hand. I find that the classical and neoclassical economists prefer to deal with continuous (convexity and smoothness) in their models. While the Marxist and post-Keynesian tend to deal with discreteness (jumps and chaos) based on dialectical reasoning.

Looked at from a methodological perspective, the matter looks paradoxical. While the leftist tend to embrace Kuhnian paradigmatic appoach in the naive sense where a new way of science completely usurp the old way, I find that they like a research program that will turn degenerating research into a progressive one. In the former group, the tendency is to appreaciate the discrete, in the latter, the tendency is to appreciate the continuous. It is paradoxical in the sense that the fall of the Former Soviet Union should indicate a break form Marxism for the Marxists. But they prefer to stay with the degenerating program, trying to make it progressive again.They seem to have one foot in the discrete domain, and the other in the continuous domain.

Dear Lall, you expand the original question. The correct answer: we need to be aware about both approaches in each model. We can skip one if the model will not become weaker. If we look at one of Hegel's law of dialectics - transformation of quantity into quality - we will clearly see the necessity of both descriptions. Like cooling water will make it an ice. Between 20 C and 10 C we should measure temperature continuously, while at t=0 C we have coexistence of two phases, like 0=ice and 1=water.

As for neoclassical economics, I would not say that they are "continuous". For prices and quantities of goods - yes. But for treatment of time and space - no. Marxists are "discrete" only because they did not elaborate math. If one would put dialectical materialism into mathematical form, there are plenty of possibilities to arrive to evolutionary models with structural change, that are using both continuous and discrete approaches.

It is more natural to describe the evolution of a system at discrete instants of time rather than continuously in time.

Source - System Modeling, CALTECH

http://www.cds.caltech.edu/~murray/courses/cds101/fa04/caltech/am04_ch2-3oct04.pdf

Dear Prof Yuri,

Complex system or non-linear discrete continuous events can be simulated and their influence on each other can be simulated with help of complex fuzzy set where amplitude can be represented their real value and phase will represent fluctuation among them at given time. Computer scientist has started simulation on this issue from 2002. Till now they are lacking in adequate measurement. It is uphold question. Really appreciate ...

I want to thank everybody for the interest to this new topic. First I will answer last comments and clarify my point of view as mathematical physicist.

Dear Prem, fuzzy set is an additional complexity that I do not consider in this discussion. Dear Arthur, you are right that there are no general mathematical rules for modelling. However, there are some given properties of nature for each case and we can simplify them if we do not lose the core of our study (see example below). Dear Krishnan, I agree that discrete programs are more useful for computer but not for the philosophical understanding.

All physicists and natural scientists know that matter exists in space and time. So first we have to study the functions of the type u(x,y,z,t), where (x,y,z) are coordinates of 3-dimensional Euclidean space (here I stay within the area of validity of Newtonian mechanics) and t is the time. Both are continuous. If we study waves in media, we can decompose space in small quasi-homogenous units and then write partial differential equations for spatio-temporal evolution of the function u. They emerge as a limit of difference schemes. This is the initial complexity. Sometimes due to symmetry we can reduce 3-dimensional problem to 2-d or 1-dimensional. We can write difference numerical scheme based on our PDE and use computer for its study. We can also move to discretization of either space or time in our theoretical model. But there are some natural limits for discretization. For example, if we study long waves with the period above several minutes, we can use discrete time model with a step about 1 minute. But if we study all spectrum, discretization will kill a fraction of spectrum.

Why am I talking about this? This is well known to those who work in the field but might be completely non-understandable to those who do not. Consider an empirical person who has some measuring devices in distance of 1 km from each other that can register time series with any frequency (almost continuous). If he would use a model of discrete space with just those points where he measures, he will ignore all parts of non-observable field for him and can miss a lot of information. A correct approach would be to use either continuous space model or with discretization given by characteristic wavelength, which can be smaller than the distance between measurement devices. It would be optimal to revise location of those devices. But to do that one should not ignore continuity of space a priori. I hope that it is clear for all physicists. Next message will be about social scientists.

Now I will talk as an economist, but will also use my knowledge of math and physics. Many social scientists never studied physics. Some of them never studied continuous mathematical methods. The link between different processes is so complex that one can hardly explain empirically more than linear relations between some variables. So if they have non-linear complex dependence in reality, this is almost never revealed in empirical model; the cloud of uncertainty is too high.

There is well developed tradition of using complex math for an interlink between prices and quantities. In abstract models an assumption of infinitely many commodities and consumers is often used. As the same time, there is no tradition to think about space and time in continuous manner, like physicists do. As for time, there may be a grounding for that. Production process takes some time, so we have an interaction between discrete points (products produced in certain days, or months, or years). This is OK, and there are even models in continuous time (see models of endogeneous economic growth).

As for space, the situation is much worse. Many economists do not have any idea about space because it is not in curriculum of economic education. Only regional science and urban economics use it, but these are special sciences (like quantum mechanics that is not needed for people who study waves of mechanical origin at macro scale). There is also a tradition to reject articles that are not based on previous literature. It happened that there was no economic literature using 2-dimensional Euclidean continuous space for economic application in the recent years.

I think (as applied mathematician) that while infinite number of spatial points and infinite number of goods are both abstractions, discretization in any of those dimensions has an equal right for existence. So I have produced (and even published few) some papers with an assumption of finite number of goods but in continuous space. Maybe somebody of physicists or applied mathematicians would have an interest for them. I guess that only very few economists have understood them so far. First I give references to my book chapter and article.

https://www.researchgate.net/publication/283908757_Role_of_Density_and_Field_in_Spatial_Economics

https://www.researchgate.net/publication/271497718_Socio-economic_influences_of_population_density

Article Socio-economic influences of population density

Chapter Role of Density and Field in Spatial Economics

Dear Yuri,

I forgot to mention that we should show some respect for the work of Robert Aumann "Markets with a continumm of Traders." Econometrica 32(1964): 39-50. He did not only claim that Perfect Competition should be modelled on a continuum, but that "when (that) notion of perfect competition is built into the model, that is, in a continuous market, one may expect that the core equals the set of equilibrium allocations." 

Dear Artur, I think that the only PDE used by many in economics is parabolic and is used to model price diffusion at stock market driven by random forces. I am not aware about any deterministic PDE. Indeed, PDE in physics often describe waves. While we also observe waves in economics (starting from business cycles and ending with long waves of Kondratiev), they have complex structure and cannot be easily described by sine waves (like some solutions of PDE).

Dear Lall, of course I know Aumann. He is not the only who has contributed to economics with mathematical ideas of infinity and continuum. For example, Dixit-Stiglitz preferences over infinity of commodities are also described by the function coming from L_p space in mathematics.

What I wanted to stress here is the lack of economic research in continuous space. There are 2 important properties of space: a) land is a measure of an area given by a subset in 2-dimensional space, b) distance is Euclidean, |A-B|=sqrt[((x(A)-x(B))^(2) +((y(A)-y(B))^(2)]. If we want to measure transport costs, they are proportional to distance (minimal, since road can have curvature), and this is missing in discrete models. Land as production factor is still rarely used in economics, and Brian Czech was talking here about its importance related to current environmental problems (partly coming from wrong accounting of the role of agriculture); see http://www.barnesandnoble.com/w/supply-shock-brian-czech/1112545943 .

Now we have approached the point where finiteness of natural resources becomes crucial. And no unlimited growth of global capital stock (by the way, not linked to any physical value, like gold or energy) via interest rate can solve it. Transportation should be also viewed not so much as the profit of transport firms, but as a tax paid to nature. If we follow the goal to minimize carbon emissions, we should also minimize it. Ideally, production and consumption should take place in the same point. This was for pre-historic people. Now we have to think again about that, facing both the growing scarcity of fossil fuels and carbon emission problem.

Dear Yuri, we actually studied discrete mathematics as a separate subject back then. I liked your remark about TV being discrete. At least in sound. For many people it's not comfortable. They don't want to be on a grid. They want to have a choice of any volume CONTINUOUSLY.  Concerning mathematical modeling, which I am doing, I would call it discrete-continuous dualism.

I would  like to generalize. It's Hegel's dialectics:  great Newton went to discretization  - finite differences, then continuous PDEs, then with the advent of computers we went back to discretization. In between was Boris Galerkin's (circa 1916) finite element method, which had a different name at the time.  General philosophical laws at work: negation of negation, and unity and competing of the opposites.

Dear Natalia, I want to clarify what I mean by discreteness in mobile phones. Before people’s direct communication was determined by their travel, which was predominantly at small distances even a century ago. Hence, most of people whom they know come from their neighbourhood (village, district, city) and this can be described as a subset of people living in their neighborhood. If I lived in point A, maybe 90% of my contacts came from the radius of 100 km around this point. When fixed phones appeared, still the majority of contacts in phone book were from this neighborhood.

Without mobile phones, I could plan whom I will call and when, and nobody could reach me when I am out of house or work. So there was a continuity in people’s plans and actions. If I agree to meet a friend at tram stop at 11:00, both will come maybe plus minus 5-10 minutes, because otherwise meeting will fail.

Mobile phone brings chaotization to our life, because; a) anybody (even 10000 km away) can call you at any moment and influence your current plans, b) people do not commit to anything, because they can call any time and tell they will not come or renegotiate.

But there are many other important points among my ideas in this post and answers that you can also discuss (as a specialist in both discrete and continuous models).

P.S. As for example of Hegel, recall his another law about transition of quantity into quality: it is essentially continuous.

Dear Yuri, I thought that the transition of quantity into quality can be best described as a step function, id est is continuous, but with a singularity.

Dear Natalia, step function is defined over a continuum of points, and is continuous in all points except for one! Discrete set is when some function is defined in points 1,2,3,4 (for example). Looking at it, one can never guess whether transition of values from the previous to the next is smooth or not. I am talking not about some discontinuities, but about neglection of a continuous set, where function is defined. Some people live in the (scientific) world where those points simply do not exist.

In your scientific application, just think that you model a planet as a discrete set of points where you got some measurements. And nothing in between!  Then all points interact in a magic way, without continuous space. You cannot imagine such thinking in geophysics, but it is common among many economists.

Dear Yuri, I did say that it is a continuous function (except for one singularity/discontinuity) defined on a continuum set.

I just missed a word "id est" in your statement. Not English? Anyway, we as mathematicians are both aware about continuity and discreteness. Some people (even scientists) are not aware about continuity. Some even do not know that there are some sets apart of several discrete points. And very often continuity is essential for understanding some properties of the world around us. This was the idea of my question.

Dear @Yuri, thanks for sharing the question. It tangles many disciplines, both the theory and practice. Regarding mathematical modelling in control engineering problems, objects/plants are rarely discrete, but mostly continuous. In general, we deal with continous time systems (objects/plants) and digital/discrete controllers.

There are two possible approach to the solution of such control problems.The first approach assume that you develop continuous controller which is then being discretized. The second approach is to discretize the continuous model of object/plant to be controlled, and then to design controller for it. Both ways bring some errors...

Hibrid control is the solution to the problem. As higher is the sampling rate for the controller, while object/plant model is continuous, the solution is is far better.

While (continuous) modell(s) and solution(s) belong to a (simple) mechanism ... machine ... installation or phenomenon, all interconnected via Large/Small Scale Onject Oriented Networks we need now/still a discrete approach. Such as we have, for now, an answer concerning the question "continuous / discrete". As fast as the non-discrete processors/computer will apare, the answer may/will be the another one.

Dear Ljubomir, thank you for support of my question and your example.

Dear Florin, I agree that networks play an increasing role in the modern world, including economics. I know about research of network topology by Duncan Watts (6 degrees of separation, etc) as well as some other works by econo-physicists. I also understand that one cannot live in the present worlds of computers without understanding discrete mathematics.

The goal of this question is to reveal an important role of continuity in thinking and modelling. If you read my other answers here, you will see many practical applications.

I think that current mathematical education and many branches of science (especially social) suffer from the lack of knowledge in continuity. Non awareness of what is continuous set [0,1] and continuous functions, only awareness of two points x=0, y=1 brings completely different thinking that is unable to deal with current challenges to our civilization. Those who are interested, please, have a look also at my other questions, about declining understanding of geography, and whether social scientists should study physics.

Some final remarks (unless there will be more interesting answers). I think that we have to admit the necessity to work with continuous-discrete dualism.

Perhaps many mathematicians know the foundations of Fourier transformation. Initially we observe a function f(x) defined on a finite continuous interval [a, b], which is defined at every point but might have finite discontinuities (like step function). Then it belongs to a class L_2 of functions, for which the integral of square in finite. There exists a countable basis of continuous functions on [a, b], like Sin (nx), n=1,2,3,..., and our function f(x)= \sum a_n Sin(nx), where a_n – coefficients, that are elements of the space l_2 (l small!), so that the sum of squares is also finite. (I am not too precise in math, but it already might seems too difficult for non-mathematicians). We are working in Hilbert space.

What is interesting? Initial function is continuous, but its spectrum is discrete (although it has infinitely many components). So this mapping of function into its spectrum has both continuity (in almost all points) and discreteness of image. Dualism, like in quantum mechanics (wave –particle).

For those who are more from engineering or experimental physics. If we observe spectrum of light from some chemical element, its components are not exact points, but have some cloud around.  (This means that we observe not a discrete, but a continuous set!) This is due to random noise in the media where a wave propagates. But normally we can distinguish between spectral components, if the noise is rather low.

When Norbert Wiener and Claude Shannon have done their theory of information, they decided to use a distinction between 0 and 1 as a bit of information. Because finite noise would not change these discrete numbers so much, that we cannot distinguish among them. Later computers were done on this basis, and this is how discreteness became more important. But remember that continuous signals are still in the basis. In the past we had analogous systems for storage of both sound and image (continuous functions), and only in the last 10-20 years digital (discrete) started to prevail. It is even used for storing photo images because with millions of pixels our eye cannot see the difference, but in reality image is initially a continuous function of space (and color, which is a wave lenghth of certain frequency in continuous spectrum of sun light).

DISCRETE...ALWAYS

So - ALWAYS? I do not want to offend you, Veena. But you are a good example of why I have put this question. New computer generation, a product of "Matrix". Hope that you have seen this movie; see https://en.wikipedia.org/wiki/The_Matrix .  People who know continuity can never be defeated by computer. I advice you to read all the posts here, and only if you understand ALL, your answer is your own, not a product of manipulation. It is not your fault, even the modern educational system is likely to be biased to more discrete thinking, than it was 30-50 years ago.

Dear Yuri! I agree with the view that the mathematical model combines the intermittent and continuous if we consider the process in analog and virtual forms.

Dear @Yuri, dear colleagues, I have read recently the experience of our colleague regarding type of modelling. 

System dynamics, Discrete Event Simulation or Agent-Based Modeling!

"For me as a modeler, the main differences between the three most commonly used methods (system dynamics (SD), discrete event simulation (DES), and agent-based modeling (ABM)) were not always clear. In my experience, the choice of the dynamic simulation model is usually done according to the previous experience of the modeler. Many times I was tempted to utilize the known and familiar DES for different problems I encountered, but other modelers suggested using SD or ABM. Recently I stumbled upon a very good series of papers, published by the International Society For Pharmacoeconomics and Outcomes Research (ISPOR), which provides an excellent overview and differentiation criteria..."

https://www.linkedin.com/pulse/system-dynamics-discrete-event-simulation-agent-based-anna?trkInfo=VSRPsearchId%3A577811291477739027460%2CVSRPtargetId%3A6087771647190777856%2CVSRPcmpt%3Aprimary&trk=vsrp_influencer_content_res_name

Yes, Ljubomir, agent based modelling is another tool (not discrete and not continuous). Some economists (Santa Fe school) use it as well. Economic agents have a complex interaction between themselves and with outer world (nature), and their actions are not determined by any kind of continuous-type field (like movement of physical elements).

I think that the problem here is in extreme complexity. Typically guys working there are good programmers, but not always good modellers. And programming is discrete by nature. Only some equations (like DE) have continuous origin, and a discrete scheme behind them does not change their nature.

But I agree that discreteness plays a higher role in social sciences comparing to natural. But it should not move out continuity completely.

Dear Yuri,

From the time when Newton formulated the fundamental anagram of calculus

http://www.mathpages.com/home/kmath414/kmath414.htm

the basic method of mathematical description of Nature has been based on the model of continuum and differential equations. The independent variables were absolute time and space coordinates. The continuous field description prevailed through the entire XIX century. The general relativity theory still retained the field description, as in the past, albeit much more complicated. The emergence of quantum mechanics complicated the picture by the wave-particle dualism allowing discrete-continuous models.

In the classical field theories the basic technique of solution of the partial differential equations is based often on the semi-discretization method. The space derivatives in equations are discretized leading to a large system of the ordinary differential equations solved subsequently with the help of an appropriate time integration method. The model is thus based on continuous description but solved for the discrete set of variables approximating the continuous variables. It is often debatable if this class of models is continuous or discrete. 

One of the best known examples of the approach outlined above is Numerical Weather Prediction addressing the flow problem in a rotating stratified fluid with condensation, chemical reactions and radiative transfer. The strict rules of convergence and consistency should be applied when solutions are verified. Unfortunately, there are very limited number of cases when rigorous analytical tests are available. This situation opens the possibility of using heuristic methods which often contain some elements of discrete representation (most often this "hybrid approach" is applied to the unresolved physical processes). 

There are also methodologies in modeling based on the assumption that the discrete systems are more appropriate for modeling of nature; the details are discussed at

http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/real.pdf

In practice, the modeling of complex systems will be based on coexistence of both continuous and discrete techniques. In all cases the main criterion of acceptability of the approach should be based on mathematical correctness (from the point of view of continuous and discrete mathematics).

Dear Janusz, you are making quite useful statement: coexistence of discrete and continuous in modelling. I think that almost all natural scientists understand that. My concern was about social scientists. Many of them have never studied mathematics of continuous, have no idea about continuous space. But they are creating models that are influencing our socio-economic life via policy implications.

Dear Yuri!

Splendid topic. Thank you very much. George Stoica delicately and correctly answered your question. Each process includes discrete and continuous components. Sometimes this is the basis for the dispute. For example, like the genius of Swift, which side begins to break an egg with a sharp or blunt end.

Dear Yuri,

Thank you for the interesting comments. I agree with your opinion concerning modeling in the areas outside of natural sciences. A notable exception is in theoretical economy and optimum control theory where the advanced mathematical methods are used routinely (for example Monge-Ampere-Kantorovich equation in the optimum transportation theory).

In the future, some models used in social sciences will evolve towards the continuous formulation with frequency distributions being the unknown functions in the field equations. We can also envision the appearance of stochastic coagulation terms, similar to those in the von Smoluchowski equation. It is also likely that the complete apparatus of theoretical physics will be applied in the social sciences, for discussion please see

http://bacon.umcs.lublin.pl/~lukasik/wp-content/uploads/2010/12/CU.Quantum.Social.Science.1107012821.pdf

Thank you very much for posting your question, it is very stimulating.

With my best regards,

Janusz

well, just about everything you do on the computer is ultimately discrete,

but in terms of ideas, that the computer cannot yet get, the continuium is

important. Problems are seldom solved,  if not by numerical methods.

Personally I prefer analytical methods.

Both paths are indispensible

There are models for materials that goes from continous to to non-continous inside ,due to increased load on boundaries or evolution of field forces , e.g. soil slope stability in my two reports where the BVP is from Pamin and deBorst (1995). Then (1996), the calculations took 2-3 days, with Matlab. The commercial codes did not follow, so no option to use them, and it is more transparent and to believe own code. (Now, it is computerised, many processes but then possibly more pointwise descriptions.)

And recent, these with another approach based on fundamental solutions  

https://www.researchgate.net/publication/319157908_Isogeometric_Boundary_Element_Analysis_of_steady_incompressible_viscous_flow_Part_1_Plane_problems

http://www.sciencedirect.com/science/article/pii/S0045782516309616

Article Isogeometric Boundary Element Analysis of steady incompressi...

Certain sets are discrete and finite in nature, for example, all accessible human population. Say that a human population is infinite could be and is incorrect.

Dear Mariano,

For systems that cannot be scaled linearily, there must be need of two properties to characterise; both cardinality and the 'space' between the numbers?

Lena, I agree that discrete objects exist in continuous space and time. Atoms are not much different from humans in this sense. If we want to study structure of humans (what biologists do), we see that body is continuous and even partial destruction of this continuity can lead to the death (example: cutting head off).

Every physicist and  biologist understands the importance of continuity. They also recognize existence of discrete objects (like apple, animal, etc), but in fact they are "discrete" only in approximation, as compact sets (continuous and bounded in space), so that one is separated from another and cannot mutually penetrate (except for sex :) ).

I have a theory for survival at the moment, and it is based on not thinking of sex and avoiding B-vitamines. (Including some exageration)

Dear Mariano, if we move from "atomistic economics", where elements are individuals (economic agents) to structural economics (see my paper below), the number of agents will become fuzzy. Because we can count families, or groups, or firms. Well, this is still a finite set, but I would be careful about its discreteness. Because agents are connected in network, and links come over (continuous) space. Cutting link in any spatial point (for example, road destruction by avalanche) deforms the network and has economic consequences.

Article Elements of Structural Economics

One of the problems is the exact nature of the connection between the two

approaches, a source of error in many calculations; because

f(x+h) - f(x) is normally subject to an infinite expansion in the form of a Taylor series

in the case of "well behaved functions".

This bothers me when I have to discretize the Schrodinger equation, for example.

I recommend the thread;

what is the name of this transfored series? is this known? In Engineering mathematics

in that the first difference of a transformed function (not the original)can be directly related to the transformed derivative function alone.(no infinite series) I think this better clarifies the nature of the connection.

I want to refresh a bit our discussion of the last year. It was correctly mentioned that observations are essentially discrete. The emergence of computers also allows only for discrete algorithms. Does it mean that we do not need continuous thinking anymore? (Perhaps some younger scientists might be asking this question).

Continuity is a philosophy. It comes from the observation of infinite divisibility of space and time. If you remember the definition of continuity (at the epsilon-delta language), it is about existence for any small number. Practically it means that some motions in space and time are smooth at any level of our measurement. It is also useful to apply differential equations derived under those assumptions to real natural processes.

"After all, a mathematical model is only a set of equations. Its link to reality is via the physical properties data of the system it is intended to simulate. Its success at simulation is not guaranteed but must be proven by suitable validation.........The basic dilemma in the process of validation may be stated in the following manner: a mathematical analogue can be validated only in a given number of known situations. Thus, no perfect validation is possible." (sic, page 238). Panjabi (1979) ...

Some would argue that model evaluation is a better term than model validation.

http://www.personal.psu.edu/faculty/j/h/jhc10/KINES574/Lecture2.pdf