Please, provide the best definition for the solution of a problem in Science, either experimental, or theoretical.
The file attached contains the manuscript af the paper, accepted for publication. The authors claim that the problem is solved, as a function, which is extremelly difficult to measure, is expressed on the basis of two other functions, easy to measure. Reviewers of an Applied Journal, refused to recognise this approach as a theoretical solution. But when the extended version of the paper, as in manuscript, was submitted for publication into more theoretically oriented Journal, reviewers recognised the derivation and the final result, as analytical solution. So, What it takes to rate the problem as being solved ?
Mathematical proof as a precursor of physical system proof is a long-standing approach to problems that are tractable theoretically but seemingly intractable physically. Strangely enough, physical proof is sometimes available but theoretical proof eludes us. For example, biologists since the time of Darwin continue to find physical evidence of the evolution of species but so far--as far as I know--mathematical proof of the evolution of the species (from microbes to humans) has not been given.
The proposed approach by @Dejenie A. Lakew makes sense, namely, finding a theoretical proof that a problem can be solved is sufficient to claim that a problem has a solution, even though a physical solution to a problem has not been given.
Dear Alexander,
thanks for sharing, but please give me some time to read the paper and built up my opinion.
Sure, Hanno, thanks for your attention. But the paper is just an example, the question is broader than this particular example.
Acceptance of a theoretical proof of a long standing scientific problem as firm and solid comes first and foremost from the person himself/herself to persistently fight, argue and present the proof as scientific and convincing, with colleague scientists and prominent scholars in the field. Proofs of such problems may not be apparent to reviewers at the moment.
We saw how painful it was for Andrew Wiles to affirm solidly and be accepted his first and subsequent claims on the proof of Fermat's last theorem of a Diophantine equation of integral solutions, who finally succeeded in convincing the mathematics community to accept his proof. It is always a matter of time and feeling comfortable with the arguments made before such proofs of longstanding problems are announced to be true and accepted as such in the field.
Another such challenging example for instance is the proof of the existence of a particle called Higgs boson, in which several international physicist work at CERN - European organization for nuclear research to observe and accept whether that particle indeed is observed and exist.
In the contrary there are cases where longstanding established and accepted theories were falsified and changed, such an example was the static universe Einstein developed a theory for it but later abandoned it and replaced with a dynamic and expanding universe.
Mathematical proof as a precursor of physical system proof is a long-standing approach to problems that are tractable theoretically but seemingly intractable physically. Strangely enough, physical proof is sometimes available but theoretical proof eludes us. For example, biologists since the time of Darwin continue to find physical evidence of the evolution of species but so far--as far as I know--mathematical proof of the evolution of the species (from microbes to humans) has not been given.
The proposed approach by @Dejenie A. Lakew makes sense, namely, finding a theoretical proof that a problem can be solved is sufficient to claim that a problem has a solution, even though a physical solution to a problem has not been given.
The examples of dear Dejenie (Higgs Boson) and of dear James (evolution) are totally convincing. I know another comparatively simple example in dosimetry. In cases where experimental result are very difficult to gain because of technical and principle problems, experiments are replaced by Monte Carlo calculations. And the dosimetrists trust in the results.
So my conclusion is, you will find an equality of theoretical mathematical results and experimental findings. And if the chances for experiments are small or impossible, I would rely on mathematical findings. The best and favourable situation of course would be the simultaneous evidence by both methods because one needs general acceptance. But that´s no scientific problem, it only handels the psychological difficulties.
PS: I don´t understand the refuse of the "applied journal reviewers".
I am with Dejenie A. Lakew… it takes ages to proof theory into practice. In mathematics for example the infinities took ages to be applied after computer invention. Another example are Newtonian laws has been formulated after many philosophers were digging into their findings such as Aristo and Galileo. These days hypotheses accumulated but if accepted then it turns to theory until it became fact and law. It also takes time.
I believe that we will never reach a state that we can claim that we solved completely a problem, because of the restriction of the incompleteness theorem. Form which we totally except that there will be truths that will never been proven... I think this will allow the scientist to be busy for over.
@Yousef,
kind hint to the Goedel theorem. But if I´m correctly informed, this theorem holds just for mathematical systems, not for the science per se. So I´m not sure, that we can apply this theorem to Alexander´s task. But I agree and hope, that scientist should not stop their research.
So long as we are on this earth, problems are bound to be there. Our research must always have an objective, once the objective is achieved we consider the problem solved. But after the outcome, we always find lope holes which, we place under recommendations. These will give another individual a stepping stone for further research, or improving on what one has done. So problems will still remain, most will be solved and many more will never be solved.
I think it all depends on the type of problem. If it is a mathematical problem, the problem is solved when the corresponding mathematical objects are found that satisfy all the conditions in the statement of the problem. The solution of the problem of finding a proof for a theorem can be considered included in the case before, because finding a proof means to find a logical argument that has the statement to be proved as its conclusion.
If it is a problem in engineering, like designing a mechanism, an electronic device, separating a chemical compound, or a piece of software, the problem is solved when the prototype is produced, tested, and found to satisfy all the specifications.
A research problem is solved when the hypothesis has been clearly stated, the experiments designed, performed, and found to confirm the hypotheses within the limits of experimental error.
The method to solve a problem is important also. In computer science we find problems that can be solved but only in principle, because of the associated complexity. There are some for which there is, theoretically, a solution which is useless in practice. A quote from Arnold Shoenberg's theory of harmony comes to my mind, connected to this:
"There is a mechanical puzzle that consists of three metal tubes of different diameters sealed in a glass-covered box. The problem is to get the smaller tubes inside the larger. Now, one can try to do it methodically,; then it usually takes quite a long time. But it can also be done another way. One can just shake the box at random until the tubes are inside another. Does that happens by chance? It looks that way, but I don't think so. For an idea lurks behind this method. Namely, that movement alone can succeed where deliberation fails."
Good answer, Oscar. My paper attached fits your defenition for a priblem being solved.
The answer, one more time, depends on the specific scientific field.
- Let's start from the easy cases, the electronic technological problems. We say that the problem of the wireless connection between a PC and a modem has been solved after IEEE had adopted the 802.11 standard: http://en.wikipedia.org/wiki/IEEE_802.11
- One way for solution confirmation is the Nobel Prize for Physics, Chemistry, Physiology or Medicine, Literature and Peace (we can omit here the Literature, since it doesn't solve problems like Alexander's original thought)
- Another popular tradition is the publication of solution from very 'difficult' journals, like Nature for example. The fact that they have accepted it means that the solution has been verified.
These are the 'legal conditions', if we want to find the essential conditions, then the question is more difficult than what it seems to be from a first sought.
Demetris provided different criteria for raithing problem as being solved. I would like to come with an example, that a Text, posted on Internet, without being accepted by any Journal, attracted attention of Mathematical Community around the globe. Russian/Jewish Mathematician from St.Petersburg provided solution for a Puankare Millenium Problem, was granted the Field Prize in Mathematics + $1 million monetary prize. But the mathematician, GRIGORIY PERELMAN, rejected both, Field Medal and $1 million. By doing so, he became a real celebraty, beyound Mathematics. He also quit his job, as a Senior Scientist in the Staklov Institute of Mathematics, survives on Ray Bread and buttermilk, which he and his mother can efford on her Old Age Security Pension. And, he claims to be happy.
In this, most unusual case, his his solution is correct, as majority of his peers rated the solution as not just correct, but being GREAT !
No Nobel Prize is needed for the solution to be confirmed. Peer revew in the Credible Journal gives sufficient assurance that the problem is solved. That is the Second Rate Journals, including those, which charge the authors for publication, can allow, either for ignorance or greed, to publish wrong claims.
Is it me, or every body feel a little bit dizzy after reading Ans Schapendonk's comments? What does she mean?
I think that a Prize is sometimes a necessary and not a sufficient condition, since by Alexander's example for Poincaré conjecture:
http://en.wikipedia.org/wiki/Poincar%C3%A9_conjecture:
the problem was solved without the Prize to be given to the Solver:
http://en.wikipedia.org/wiki/Grigori_Perelman
Thus the general agreement for the solution among the specialists of the relevant field is enough.
But, what if a general agreement exists for a 'solution' and after more than 100 years the validation of the thoery never comes? This is the hot point...
Good point about GENERAL AGREEMENT, Demetris. As for the theory, no one ever derived Maxwell equations for electromagnetic field, but they are accepted, becouse completely validated experimentally, and GENERAL AGREEMENT that they are correct, and, the problem is solved.
I think that we could distinguish between theoretical mathematical problems and experimentally tested natural scientific problems. In fact, in an axiomatic system like maths, it is feasible in principle to get an answer by simple thought arguments, but in physics for example this way is not a complete one: Your thoughts that have led to a theory must be certified by the experiment at least in greater accuracy than the precious existed theory on hand. Another fact that makes situation harder in natural sciences is the commonly accepted view that a theory can be dropped down only by one only contradictory example.
But, this is not how people really work in science: In most of the cases the accuracy with a % of available data is enough for the general consensus. Even Maxwell's EM theory cannot explain some phenomena, but, since they are a small % of the total, we 'do not concern' about this lack...
If we wanted to be 100% accurate then most of our 'holy theories' wouldn't had passed the tests...
Maxwell Equations work perfectly in the range within long radio-waves to visuable light. At shorter wave length quantum effets appear, Quantum Ekectrodynamis takes care of them. Maxwell Equations are the first Relativictic Equations in the History of Science. They are invariant to Lorentz transform group, which Einstain later derived within Special Relativity.
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