Some one has rightly said` God has created universe and Man has made Mathematics to understand it ( my be approximately)!

( Man should be understood : the man kind including male and female)

Not all problems can profitably be mathematicized. Sometimes descriptions are better.

For example there is very little theoretical biology. You do not need math to describe the structure of a cell.

If you study the atomic structure, you first mostly need a description

of the nucleous in terms of protons and neutrons, and then with the same number of electrons as protons,

with respective sizes, masses, distances. Only later maybe get into quantum math.

Hi, I agree with Juan's answer. Physical problems modeled by a system of ODEs or PDEs can benefit from mathematics for example. It is also the case for geometrical problems.

Mathematical modeling often seems affordable. But this is just a tool for a good head. Often a good tool and a bright head do not meet each other. In this case we observe the birth of a large number of surrogate models that are not capable of solving practical problems. Often, in pursuit of spectacular modeling, the real physics of the process is sacrificed. As a result, we have a kingdom of curved mirrors.

Adding a different aspect to the usefulness and significance of mathematical models in physical sciences, it is worth mentioning that, sometimes, they open up new and unexpected perspectives, as it happened in the early ‘70s, when simple quadratic recurrence equations that biologists used, at that time, to model population growth in some species, helped to reveal fundamental properties in non-linear systems, such as the Feigenbaum constants.

Ничего нельэя абсолютизировать. Но, говоря о машинах великий русский поэт сказал:

Здесь что? Мысль роль мечты играла,

Металл ей дал пустой рельеф

Смысл — там, где змеи интеграла

Меж цифр и букв, меж d и f!

В. Брюсов

В. Брюсов еще написал: "Пред волей числ мы все – рабы."

Суррогатные модели делают нас такими рабами.

Generally Each problem has a solution but not all the time

Every attempt to understand an aspect of the real world begins with a more-or-less accurate and precise description of the relevant situation and deduces conclusions from that. This description need not be in mathematical form, but such a form has the advantage that one can count reliably on the mathematical manipulations used for the deductions involved, including analysis and simulation. The disadvantages come from the necessity to interpret the mathematical conclusions and, of course, the difficulty of assessing the accuracy of the mathematical description and the precision of the measurements involved. What is typical here is that in creating a mathematical model one approximates because the model would otherwise be too difficult to work with or because one ignores unfamiliar phenomena which do not occur in more familiar contexts.

Some one has rightly said` God has created universe and Man has made Mathematics to understand it ( my be approximately)!

( Man should be understood : the man kind including male and female)

There are two opposing approaches to research. First: write out beautiful equations and try to prove that nature behaves in accordance with these equations.The mathematicians is often working by this way. Second: examine the totality of known facts and try to establish the correct patterns. This way the physicists are working.

My opinion is that the second approach is more true.

The working model should be based on facts, no matter how paradoxical they may seem. It is of course ideal when the mathematicians help to physicists.

Note. This thought from the book of my longtime colleague and I would recommend it with pleasure to read https://www.amazon.com/Physical-Space-World-Valeriy-Polulyakh-ebook/dp/B008NBGTOO

To Amia K. Pani (Some one has rightly said` God has created universe and Man has made Mathematics to understand it.)

Attempting to put mathematicians on par with God looks funny. Behind it lies the banal promotion of their business interests. It is easy to make sure of it. Just try to create the simplest theory of what you regularly observe in your garden - for example, the turning of a small grain into a flowering shrub.

A few quotes from the book I mentioned ( https://www.amazon.com/Physical-Space-World-Valeriy-Polulyakh-ebook/dp/B008NBGTOO )

Periodically local crises in the science emerged. A very serious crisis came in physics during the late 19th and early 20th centuries. As a result, a special theory of relativity and quantum mechanics came to life. However the science acquired methods of calculations of measurable parameters at the expense of physical clarity. There is a famous statement of Richard Feynman: "I think I can safely say that nobody understands quantum mechanics." From this time physics becomes more and more "non-obvious". Mathematics devoured the remnants of physical reasoning. A manifest example is a string theory; mathematicians think that this is physics but physicists think this is mathematics. And no one knows which entity is circumscribed by this theory. A stream of illusory mathematical constructions of "reality" is inexhaustible, going continuously into the sands of oblivion.

Without profound physical reasoning it becomes simply mathematical pyrotechnics which produces bright sudden outbursts which burn down without considerable consequences. I want to be clear; although the physics substantially based upon mathematics, however it is not reduced to mathematics only, otherwise what difference would there be? A mathematical model is not a substitute for reality. Mathematics is only the tool of physics.

A predominant instruction of the senior researchers to their young colleagues is "Shut up and calculate". Any reasoning beyond the formulae is prohibited. Models are founded usually on the specific mathematical knowledge of practitioners, on their background and sometimes do not have any relation to reality. However this approach contradicts the normal path followed by the previous generations of scientists. On this path the physical verbal logical reasoning have to give way to formulae. A physical hypothesis that is following from the observations qualitatively ought to be cast in the formulae to become a quantitative theory. It is not productive to attribute beautiful formulae to the reality without profound physical reasoning. Unfortunately, it is fashionable now to believe that the physics is a naked formula, not a system of physical ideas. In this way the physics can go too far into abstractions and a comeback from mathematical jungles to reality would be too painful.

Leopold Infeld in his "Quest" recalls distinctly his talks with Einstein: "His point was that there are amazingly few fundamental ideas in physics and they can be represented in words. As he said: "No scientist think in formulae." Before a physicist begins to calculate he must have some picture or some pattern of reasoning in his mind which can, in most cases, be formulated in simple words. Calculations and formulae are the next step."

I think Rinat Seidgazov would appreciate https://www.basicbooks.com/titles/sabine-hossenfelder/lost-in-math which argues against the way Mathematics is currently being used in Physics --- but Hassenfelder's complaint is quite different from his, although in each case the criticism is addressed to Physicists, not to Mathematicians. He argues that they should rely on the ideas (amounting to well-formulated intuition) where she argues that the criterion for formulas has lost contact with experiment. The notions of "verification" are important for each field, but are different and are important for different reasons.

It is also possible for Mathematicians to grind out formulas without asking what they "really mean" and, as with Physics, this sometimes leads to important understanding later (one example is the purely formal computations with complex numbers used originally for intermediate steps in solving cubic equations to get the correct real answers; another is the introduction of the Dirac delta function, later justified by Schwartz). The importance of the "ideas behind the formulas" lies in developing good intuition to know which formulas to use when: Mathematics always has assumptions and one can use the formula only in the context of those assumptions (an example is the ridiculous formula 1+2+4+8...=-1, using the geometric series for 1/(1-x) with x=2 when the assumption is that |x|

Many thanks to Thomas I. Seidman for recommending the book.

I would not like to divert the discussion in the abstractions of the universe and social psychology.

A few decades ago, it just seemed to welders that they had studied the physics of the beam welding process quite well and firmly held God by the beard. With the availability of computer equipment and universal software, there was a temptation to abandon complex physical studies.

Mathematicians took up the case, ignoring the work of analyzing experimental data and verifying hypotheses (after all, they need just a quick result with beautiful graphics). The numerical experiment completely superseded the real measurements. As a result, there are hundreds of computer models of welding now, none of which reflect the real hydrodynamic processes and therefore are not able to solve practical problems (except for that narrow problem, under which it was specially tuned with “crutches” of fitting coefficients).

It is widely believed that in the field of industrial technology, any hypothesis and the model resulting from it are reliably verified by experimental data. But it turned out that a catastrophic situation has developed in this area with the verification of the hydrodynamic model of welding. Moreover, the depth of the disaster is not yet realized by the welding community.

Let me give an example which is more familiar to me (as a former Boeing Mathematician) than the hydrodynamic model of welding. In the days of the Wright brothers, testing a new plane design was done by sending up a test pilot. Theory and experiment suggested the Reynolds number scaling which permitted companies to get useful information with (physical) models in a wind tunnel and singular perturbation analysis (boundary layers) to understand why the aerodynamics worked as it did. One major advantage (besides keeping test pilots alive) was that it was cheaper and faster to build such models and vary details of the designs (and measure wind flows) than to build full-size prototypes and the state-of-the-art shifted over to wind tunnel models (and some analysis including conjectures like the "area rule") with prototypes only for a few examples.

With the development of modern (large, fast) computers and the evolution of numerical analysis the same history repeated for the same reasons: it is again cheaper and faster (and arguably more accurate and informative) to compute rather than to use physical models. For supersonic regimes, the use of physical models may not be possible at all.

I will conjecture that it will be impossible to find industrial welders with well-developed intuition (by experience) in unfamiliar settings such as under water or in vacuum or at very small scales or where hazardous environments make it desirable to control remotely or ... At that point it would be unavoidable to work with analytic models -- with a necessity for verification, although there would certainly be a division of labor between the analysts and the verifiers.

In vain you consider me an opponent of numerical simulation. It's not like that at all. At the beginning of my career, I myself did numerical calculations. But each stage of the calculation had to be checked with experiments.

Today, as a physicist, I wonder why mathematicians allow themselves to ignore the facts fairly easily. Is such a drop in qualifying requirements?

I do not understand why you oppose the computational model and the physical model? After all, it must be the same thing, only described by different means. Ideally, a calculational model should always be based on a physical model. And only in the case of large computational difficulties, engineering concessions in the form of approximation models are allowed. But such models are not accidentally called surrogate.

I believe that good avionics does not appear where somebody are able to make beautiful graphics (although this is not bad), and where modern wind tunnels work, there are qualified physicists and tight control of calculations. I think that Boeing flies for this reason.

To answer this we should first be able to define mathematics; if we take mathematics the what ZF+AC brought us, maybe not every thing.

If we consider other foundations for mathematics, maybe.

Taking all to gather, in my view, this question can never been answered.

Of course it can be mathematically modelled but the effeciency in the real world will fluctuate. Despite of such obstacles.... mathematical models have proved worthy in many aspects of medicine, biology, physics, etc.

Thanks to all.

everything could be modeled but the limit is creating a lively thing with blood and meat (creating and not modifying )

some things could be modeled and understood ,such as blood cells for example ,but technology can't build a real blood cells from scratch ,only artificiel

Math has a bit of everything in it, abstraction, logic, intuition, calculation, numbers, modeling, sets; some people are addicted, others scared.

Really, impossible to define.

To Rinat Seidgazov

On `Attempting to put mathematicians on par with God looks funny. Behind it lies the banal promotion of their business interests. It is easy to make sure of it. Just try to create the simplest theory of what you regularly observe in your garden - for example, the turning of a small grain into a flowering shrub'

Possibly you have misunderstood the comment. It is never intended to put --- to par with God.

In that case, excuse me for some emotionality. Although it is impossible to work without emotions.

Yes because God is the Greatest Mathematician :-D

You're right. God is such a brilliant mathematician that it is impossible to present such a committee that decides on the Award to such a Laureate :)

Why not?!?!!? The real question is whether you will be satisfied with the results or the outcome the final model suggests? I'm afraid that scientist (or human as we all are) would probably find a way to justify or by pass suggested results, regardless of its usefulness.

I think to modeling a phenomenon you must understand and describe this phenomenon first..if you do... you can translate this phenomenon to a mathematical model.

Till this day...is there a phenomene wich can be understoded and can't be modeled ?

Yes of course we can describe phenomenon using mathematical equation. In fact in some sense equation already exits, just we need to identify behavior of equation for particular phenomena. There are atmospheric optical phenomena which is not mathematically modeled.

Yes, I think that the answer for this is almost positive. If the all information about the phenomenon is known, then model can be proposed. Models are to answer some special kind of questions, so better not to expect too much from it. We can not find all answers in one model that is why no model is perfect.

Modeling of a phenomena depend on our interpretation of it. So, we just can model things in our thought and language, which make mathematics.

The development of applied mathematics and specially optimaition theory and the formulation of mathematical models also what we see today from the enlargements applied to many areas in our lives suggest to us that the manufacture of mathematical modeling will invade all walks of practical life, so I agree with the view that every problem in life can be transformed in to mathematical model and it remains only how to solve that model.

Let us answer it indirectly by asking the question: "How many naturally observed phenomena are described by the mathematics in biology and medicine?"

The honest answer is: "Surprisingly little!"

The reasons are multiple. The main one is caused by a lack of relevant theories capable of precisely describe complex systems and carry on their quantitatively correct modeling and prediction.

This leads to the situation that -- currently -- most of the biomedical research is describing the effects of various types of interventions on their behavior without being capable to do it from the first principles.

Nevertheless, new emerging approaches are being developed, which are based on complex systems and their mathematical apparatus, that enables to predict of the evolution of the single biological system under consideration instead of performing only statistical analysis of a large number of similar cases.

Complex systems models & future theories give us for the first time capability to predict

* Brain aneurysm pressures from CT scans without using invasive probes!

* Hydrodynamics of the heart.

* Evaluation of interaction of blood flow with arterial walls.

* Detailed spatiotemporal description of the spread of diseases in societies and in living tissues.

* Agent-based modeling of carcinoma.

* Use of complex systems in the prediction and treatment of cancer resistance development.

* AI and machine learning techniques used to scan huge databases in search of hidden interdependencies.

* Complex systems entropy measures enabling to asses various modes of ECG and EEG recordings and hence to detect various diseases and even to predict them in some cases.

There phenomena that cannot be modeled, for example in economics and social sciences.

Любой абсолютно однозначный ответ на такие вопросы всегда не верен. В том числе и данный.

The key is symmetry

It depends on what the word "phenomena" means. Can you mathematically model "mathematical phenomena"?

Sometimes you do not model, you count money to be able

to reach the end of the month. Is that modeling?

Agree with Ivan.

The question as to whether a "phenomenon can be mathematically modeled" is really two questions: (1) Is there a possible mathematical description of this phenomenon? and (2) Can we (easily?!) find such a description?

The question (1) does not ask whether we already know such a description, but I believe that, for any phenomenon we understand, that understanding would necessarily involve some sort of structure --- which can then be classified as "Mathematics."

The question (2) addresses the state of our knowledge and the mathematical tools (whether analytical or computational) which are available as well as the data needed to determine/estimate whatever parameters might occur in the model. Perhaps the major concern here might be the range of circumstances one wishes to consider and the accuracy desired, since all feasible models necessarily ignore aspects labeled as "negligible."

I belive so.

The question of what can or cannot be (usefully) mathematically modeled is very deep. Some years ago, I tried to tackle it in the context of mathematical modelling of only "Life Support Systems". If anyone is interested, there is a published (largely forgotten, non-technical) article from 2002.