Science is special kind of human knowledge, human achievements, and human activity related to elaboration of the specific language, id est, system of concepts and notations for usable description of reproducible phenomena of any origin, characterized in the following:

[1] Applicability. The language is built up of the concepts. Each concept has limited range of applicability. For example, concepts that operate with "theory of everything" or "set of all possible sets" are not allowed.

[2] Verifiability. Each concept allows the verification: Within the terms of already accepted language, some experiment with some specific result, that confirms the concept, can be described.

[3] Refutability (negability). Each concept allows the negation: Within terms of this concept, some experiment with some specific result, that negates the concept, can be described.

[4] Self-consistence. No internal contradiction for the concept are known.

[5] Correspondence. If the area of applicability of a new concept evolves that of another already established and verified concept, then, in this area, the new concept either reproduces the results of the already verified concept, or indicate a way to negate the previously established concept.

[6] Pluralism and simplicity. Co-existence of mutually-contradictive concepts, satisfying criteria [0-4] above is allowed. If two concepts satisfying S1-S5 have some common range of validity, then, in this range, the simplest of them has priority.

The two last reqiorements specify the relations of a sciencific concept to other scientific concepts. They together can be called tolerance. Such tolerance means that the concept somehow respects other concepts.

The 6 declarations above were used as axioms to construct site TORI, destroyed 2013.02.27 by Nishio Shigeaki; then, 2013.03.06, his vandalism had been approved by Hitoki Yoneda, Akira Shirakawa, Mitsuru Musha, Takashi Mukaiyama, Hajime Nishioka, Ken-ichi Nakagawa, Nobuyuki Nakamura, and Makoto Morinaga; see http://budclub.ru/k/kuznecow_d_j/toriattacked.shtml.

Structure of science

The scientific concepts are constructed on the base of observations, experiments, definitions, axioms, theorems, hypothesis and theories.

Observations are related with identification of similarities in the events of some class. For example, one may observe that the same rules of counting are efficient being applied to objects of different origin: stones, bottles, humans.

If the observation requires some action from the researcher, it is called experiment. For example one count different objects just in order to verify the observation mentioned in the example above.

Definitions are used to make the description shorter, operating with new terms, created for some specific concepts. For example, one may define the natural numbers to simplify counting of some objects that are considered in some sense equivalent.

Axioms are statements that are assumed for some concept or some paradigm.

Theorems are statements that deduced on the base of axioms and definitions.

Hypothesis are statements that are formulated within some paradigm, concept and are expected to be either proven or negated by the deduction or by some observations or experiments. While the hypothesis is negated, it is not supposed to be used.

It the hypothesis is proven by the deduction, it becomes theorem.
If the hypothesis used to predict some non-trivial result of the experiment and the experiment confirms it, it becomes theory.
According to the property [5], several mutually-contradicting theories are allowed, until some of them are negated either by deduction or by experiment.

The properties above are supposed to be common for all sciences. The job, work related with making a science is called research; although research may be considered also as some specific, "small" kind of some more general science.

Classification of sciences

Sciences are classified by several criteria

Natural versus humanitarian

The main specification, invented by humans, specifies, does the science apply to themselves or not.

Natural science is any science that can be formulated without use of concepts related to human, humanity or human society.

Humanitarian science is any science that cannot be formulated without the concept of human, humanity, or the human society.

In principle, one researcher can make a dirty trick with the definitions above. Some science may deal with some societies (population of self-motivated robots, ants, bees, monkeys, etc.) without to specify their origin, and this science should be qualified as natural. If one day the researcher finds that some elements of his science (observations, experiments, etc.) can be applied to humans and are specific for humans, the science becomes humanitarian.
Similarly, while one researches about a medicine, drug or a weapon which is supposed to be applied to animals, it is natural science; but as soon as human appears on the science (for example, a farmer who is interested to cure or, contrary, to kill his domestic animals in the most easy, safe and efficient way; or a geophysicist, who makes some research far from civilization and needs to have a gun ready to protect himself, or a hunter), the science becomes humanitarian.
This classification holds even for completely anti-humanistic, barbarian case, aiming the elaboration of a weapon of mass destruction, zombing, etcetera; formally, the such a science remains humanitarian.

Applied versus fundamental

Any humanitarian sciences or any natural science can be qualified as
applied, if the research is expected to bring some specific advantage, for example, economical benefit, or can be used in other research; and fundamental if the main expected application is difficult to specify.

For example, the analysis of quantum fluctuations of topology in vicinity of vaporizing blackhole should be considered as fundamental science, and the optimization of the design of a car engine is applied science. If the science has narrow area of application, then the word "research" can used as noon instead of science. In this sense, one may say, that optimization of a specific engine for a specific car is not a science, but still research.

Experimental versus theoric

Also, any (humanitarian or natural, and fundamental or applied) science or research can be classified by the method used. If the scientific results are obtained mainly as a result of deduction, then the science (and the research) is called theoric; and if experiments dominate, then the science is called experimental. There is no abrupt gradation between these two kinds of science; often, the research implies both, the theoretic analysis, deduction and the experiment.
Also, some intermediate position is occupied by the numerical simulations; sometimes such a simulation is called "numerical experiment".
However, the gedanken experiment still remains as tool of the theoretical science, not the experimental one.

Justification of science

Usually the science does not give the immediate benefits. The most of science is realized for funds from the governments of civilized countries, from phylantropists and from the advanced companies that use the modern achievements of science to remain at the top level of technology.

foundation of science

The distribution of funds assigned for science is always a serious problem. In the most of cases, the administrators have no abilities to drill deeply into the research they are funding. Therefore, the financial support is distributed on the base of formal criteria: publications, citations, participation on conferences, regular character of the scientific seminars. The skills to write the grant applications and ability to keep good relations with colleagues and administrators are very important, if not dominant factors in the getting foundation for the scientific research. For the same reason, the spectacular character of the new effects achieved helps a lot.

Especially poor distribution of funds occurs in the countries with high level of corruption; the significant part of the foundation does not reach the destination. Also, in such countries, the government tends to keep some scientific achievements in a secret, for the application in the military industry. In the most of cases, such achievements are fake, because they are protected against the critics, and the errors, mistakes, have no opportunities to be revealed at the early stage.
Also, in totalitaristic countries, some branches of science may happen to be suppressed just for some diplomatic reasons with demagogic pretexts. For example, it had happened in the USSR with the Theory of Relativity (even the special relativity), Quantum Mechanics, Cybernetics, Genetics.

Hierarchy of science

At the level of 20 century, the following hierarchy of sciences is accepted:

The top of science is fundamental mathematics. No one other science contradicts mathematics. However, in principle, different formalisms within mathematics are allowed compete.

Mathematics is considered as a base of science, top of the human knowledge. Language of mathematics is used in all other sciences. No one other science dares to contradicts mathematics.

Computational mathematics and Computational science appeared as some intermediate step, a layer, between Mathematics and other sciences; initially only between mathematics and physics. Only at the end of XX century this "layer" takes its shape as a specific and partially independent science; but all its statements and concepts are supposed to agree with basic mathematics.
In computational mathematics, the mathematical statements statements can be verified with hundreds decimal digits. This can be considered as some kind of verification of the mathematical axioms; most of humans have no doubts in these axioms.

General and fundamental physics can be considered as some bridge from mathematics and computational mathematics to all other human reality. All physics is based on mathematics. Fundamental concepts of physics use some small parameters that allow to refine and to perform the measurement, and postulate the laws of Physics in terms of observable quantities, or simply, observable; such laws are formulated mathematically and hold with tens decimal digits; the best precision reported is supposed to increase from year to year. However, since century XX, the fundamental physics operates also non-measurable quantities. Some religious activists and even researchers used to criticize the Quantum Mechanics for using non-observable quantities. In Century XXI, the most of researchers accept the use of non-observables quantities, because such description leads to the most precise measurements of the observables even reported.

Chemistry sometimes is considered as a special science, especially while the capacities of quantum mechanics and computational mathematics do not allow to calculate all the properties of chemical compositions numerically.

Something-physics, like geophysics, astrophysics, hydrophysics, glaciology, etcetea sometimes are considered as special sciences; as well as various technologies and industrial sciences.

Biology deals with self-reproducing objects that cannot yet be created by methods of physics and mathematics. Such objects are called "living", life. The methods of biology, genetics and selection allow slightly modify the type of biological objects, but not yet create them from non-living matter. The hypothesis about the creature of primary life cannot yet be answered within the science and belongs to the area of Theology, religion. Many claims to deduce some statements about the primary life were reported, but they do not satisfy some of criteria (0-6) of the scientific concepts.
The science about description of biological objects in terms of physics is called biophysics, although sometimes this science uses also biological terms and concepts that have no equivalent in physics. Similarly, the description of complicated organic molecules with terms of physics and chemistry is related to bio-chemistry.

Medicine can be considered as a part of biology, but it deal with human and therefore, according to the definition above, it is humanitarian science. Medicine has specific feature: in the most of cases, the direct experiments in Medicine are prohibited by Law.

Other humanitarian sciences are related with behavior of humans, and specific objects and concepts closely related with human activity: History, Economics, Sociology, Literature and linguistics, Artistic science, Law, Politics. In this raw, each sequent science at least pretends to agree with the previous ones. No one science is supposed to contradict the physics, The Sociology and Economics try not to contradict the History; Law and Economics try not to contradict each other, and politics always pretend to be based on the History, the Economics and the Law. These sciences are not yet well developed and a lot of mutual contradictions are under discussion.

Human knowledge

The human knowledge is not limited by the science. A lot of achievements of technology and culture, based on human experiences does not match the criteria formulated above.

Entanglement of the human knowledge

The human knowledge is highly entangled.
The history implies that, there are associated customs and skills, that allow humans to survive in a certain historic epoch.
The economics implies that there exist business.
The literature and linguistics have no sense without people who write, read, or, at least, talk to each other. Politics implies that some people in the society have some common interests, that allow them to share the information, to discuss and to find some compromises, although many cases of misinformation and even wars are currently under discussion.

Religion

The religion takes the special role in the human knowledge. Most of achievements and tragedies of human civilizations and associated with some religion.
the Arabian civilization, that brought the humankind the fundamentals of Al-Gebra, was based on the Islam.
The so-called Western civilization, that brought the most of technical achievements, used in IXX and XX centuries (steam machines, cars, aircrafts, opto-electric devices, computers) were based mainly on the Judaism and the Christianity, although the achievement of the Roman Empire in the Science of Law (used until now) seem to be related with the Polytheism.
The philosophy of pacifism, that allows the human civilizations to avoid at least big military conflicts already during more than half-century (touch a wood!), came from countries, where the dominant religion is Hinduism.
Religion-less societies almost did not exist in the human history. There were few intents to build-up a society that exclude the religion. Such communities ether did not last long, as the Paris Commune, or led to creation of a new religion, for example, the Marxism-Leninism, as it happened in the Soviet Union. This indicate that religions should be considered as important part of the human knowledge.
Many religions have some of elements, typical for the Science. However, all the religions fail the refutability. In addition, many of them (like Islam, Judaism, Christianity, Marxism-Leninism) fail also the pluralism, required for the scientific knowledge.

Conclusion

There are many references related with topic Science, and it is difficult to choose the appropriate ones. In addition, I disagree the definitions of Science I meet in the literature. This is reason why I wrote this article. Below, I mention only few references, which are not most reliable sources, but most general and popular.
http://en.citizendium.org/wiki/Science
http://en.wiktionary.org/wiki/science
http://en.wikipedia.org/wiki/Science


_{External links:
http://www2.lse.ac.uk/philosophy/About/lakatos/scienceAndPseudoscienceTranscript.aspx Appraising Lakatos. Science and Pseudoscience. (1975)
http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html Eugene Wigner. The Unreasonable Effectiveness of Mathematics in the Natural Sciences. Communications in Pure and Applied Mathematics, vol. 13, No. I (February 1960).
https://docs.google.com/viewer?a=v&pid=explorer&chrome=true&srcid=19oUIxa5S5hhF53M7BY33fi07AtiOcRodVV26vW3FRfR7Du621cGWFUK2XkIv&hl=en Ralph Leighton. "Surely You're Joking, Mr. Feynman!"
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Copyleft 2010 by Dmitrii Kouznetsov. This text may be used for free, attribute http://budclub.ru/k/kuznecow_d_j/science.shtml}