Sciences.ch (introduction)

No human endeavor has had more impact than the science on our lives and our conception of the world and ourselves. His theories, his conquests and results are all around us.

Omnipresent in industry (aerospace, imaging, cryptography, transportation, chemistry, ...) or in the services (banking, insurance, human resources, projects, logistics, architecture, tcommunications, ...), applied mathematics also appears in many other areas: surveys, risk modeling, data protection ... They influence our lives (telecommunications, transport, medicine, meteorology, music ...) and contribute to the resolution of current issues: energy, health, environment, climate, optimization, sustainable development ... They great success are their fabolous dispersion in the real world and their increasing integration in all human activities. We are going therefore to a situation where mathematics will no longer have the monopoly of mathematics, but where economists, managers and commercials will all do mathematics.

As a former student in the field of engineering, I have often regretted the absence of a single book fairly comprehensive, detailed (without going to the extreme ...) and educational if possible free (!) and portable (being personally a fan of electronic reading ...) containing at least a non exhaustive idea of the overall program of applied mathematics in engineering schools with an overview of what is used for real in companies with more intuitive than rigorous proofs but with enough details to avoid unnecessary effort to the reader. Also a book that does not require to adapt each time a new notation or terminology specific to the author when it is not outright to change to a a foreign language ... and where anyone can suggest improvements or additions.

I was also frustrated during my studies to have quite often have to swallow "formulas" or "laws" supposedly (and wrongly) unprovable or too complicated as my teachers says or even disappointed by renowned authors books (where developments which are left to the reader or as exercise...). On this website and the associated PDF, predominates the will to never confuse the reader with empty sentences like "it is evident that ...", "it is easy to proove that ...", "we leave it to the reader as an exercise... ", since all developments are presented in detail. But I'm not a purist of maths! I have only one ambition: to explain the easiest way possible.

Although I have to admit that proove some relationships presented within the engineering schools curriculum can not be done because of a lack of time in the official program or size limit in a book, I can not accept that a teacher or author tells his student (respectively, the reader) that certain laws are unprovable (because most of the time this is not true) or that such or such proof is too complicated without giving a reference (where the student can find the information necessary to satisfy his curiosity) or at least a simplified but satisfactory proof.

Moreover, I think that it is totally archaic today that some teachers continue to ask to their students to take a massive a amount of notes. It would be much more favorable and optimal to distribute a course handout containing all the details in order to be able to concentrate on the essentials points with students, that is to say the oral explanation, interpretation, understanding, reasoning and practice rather than excessive blackboard copy... Obviously by giving a complete course handout some students will be brilliant by their absence but ... it is the better! Thus, those who are passionate can deepen subjects at home or at the university library, the weak do what they have to do and the rest (struggling students but workers) will follow the course given by the teacher to profit to ask questions rather than mindlessly copying a blackboard.

Inspired on a learning model of an American scholar, whose I forgot the name ... this website (and its associated PDF) proposes and imposes the following properties to the reader: discover, memorize, cite, integrate, explain, restate, infer, select, use, decompose, compare, interpret, judge, argue, model, develop, create, search, reasoning, develop in a clear progressive teaching way to develop the analytical skills and openness.

So, in my mind, this website (and its associated PDF) must be a substitute, free of charge, to many references and gaps of the system, allowing any curious student not to be frustrated for many years during his school education. Otherwise, the science of the engineer coule have the aspect of a frozen science, apart from the scientific and technical developments, a heteroclit accumulation of knowledge and especially of formulas which made he considered as a tasteless subproduct of mathematics and that brings companies to many false results ...

Those who see applied mathematics only as a tool (what it also is), or as the enemy of religious beliefs, or as a boring school field school, are legion. However, it is perhaps useful to recall that, as Galileo said, "the book of nature is written in the language of mathematics" (without wishing to do scientism!). It is in this spirit that this website (and its associated PDF) discusses applied mathematics for students in the Natural, Earth and Life sciences, as well as for all those who have an occupation related to the various subjects including philosophy or for anyone curious to learn about the involvement of science in everyday life.

The choice to study engineering on this website as a branch of applied mathematics comes from the fact that the differences between all areas of physics (formerly known as "natural philosophy") and mathematics ars so hardly notable that Fields medal (the highest award today in the field of mathematics) was awarded in 1990 to physicist Edward Witten, who used physical ideas to proove a mathematical theorem. This trend is certainly not fortuitous, because we can observe that all science, since it seeks to achieve a more detailed understanding of the subject it studies, is always finish his trials in the pure mathematics (the absolute path by excellence ...). Thus, we can predict in a far future, the convergence of all the sciences (pure, exact or social) to the mathematics for the modelisation technics (see for example the PDF "L'explosion des mathématiques" available in the Download section of the website).

It can sometimes seem to us difficult (due to irrational as obscure and unjustified fear of pure sciences in a large fraction of our contemporaries) to transmit the feeling of the mathematical beauty of nature, its deepest harmony and the well-oiled mechanics of the Universe, to those who know only the basics of algebra. The physicist R. Feynman spoke a day of "two cultures": people who have and those who do not have sufficient understanding of mathematics to appreciate the scientific structure of nature. It is a pity that mathematics are necessary to deeply understand nature and that they also have a bad reputation. For the record, it is claimed that a King who asked Euclid to teach him geometry complained about its difficulty. Euclid replied, "There is no royal road". Physicists and mathematicians can not convert themself to a different language. If you want to learn about nature, to appreciate its true value, you must understand its language. The nature is revealed only in this form and we can not be pretentious to the point of asking him to change this fact.

In the same way, no intellectual discussion will allow you to communicate with a deaf person what you really feel while listening music. Similarly, all discussion of the world remain powerless to transmit an intimate understanding of the nature of those of the "other culture". Philosophers and theologians may try to give you qualitative ideas about the Universe. The fact that the scientific method (in the full sense of the term) can not convince the world of its truth and purity, is perhaps the fact of the limited horizon of some people who imagine that the human or another intuitive concept, sentimental or arbitrarily is the center of the Universe (anthropocentric principle).

Of course, in order to share this mathematical knowledge, it is paradoxical to increase, with our work, the long list of books already available in libraries, in commerce and on the Internet. Nevertheless, we must be able to present a strong argument that justifies the creation of such a site (and its associated PDF) as compared to books such as Feynman, Landau or Bourbaki. Here are some arguments:

And also ... I believe that the results of individual research are the property of humanity and should be available to all those who explore anywhere the phenomena of nature. In this way the work of each benefit to all, and that is for all humanity that our knowledge cumulates and this is the trend that allows Internet.

I do not hide that my contribution is limited largely to this day to that of a collector who gleans his information in the works of masters or publications or from anonymous web pages and who completes and argues developments and improved them when this is possible. For those who would accuse me of plagiarism, they should think on the fact that the theorems presented in most non-free books and commercially available have been discovered and written by their predecessors and their own personal contribution was also made, like mine, to put all this information in a clear and modern form a few hundred years later. In addition, it can be seen as doubtful that we ask to pay for access to a culture that is certainly the only truly valid and fair one in this world and where there is no patent or intellectual property rights.

This website (and its associated PDF) also reflects my own intellectual limitations. Although I try to study as much science and math fields as possible, it is impossible to master them all. The website (and its associated PDF) shows clearly only my own interests and experiences as consultant, but also my strengths and my weaknesses. I am responsible for the selection of inputs and, of course, of possible errors and imperfections.

After attempting a strict (linear) order of presentation of the subject, I decided to arrange this website (and its associated PDF) in a more pedagogical (thematic) way and always with practical examples o applications. It is in my opinion very difficult to speak of so vast subject in a purely mathematical order in only one human life, that is to say, when the concepts are introduced one by one, from those already known (where each theory, operator, tools, etc.. would not appear before its definition). Such a plan would require cutting the website (and its associated PDF), in pieces that are not more thematic. So I decided to present things in a logical order and not in order of need. Thus the reader will encounter, as the editor himself, to the extreme complexity of the subject.

Some know that for every theorem and mathematical model, there are almost always several methods of proofs. I've always tried to choose the one that seemed the most simple (eg in relativity and quantum physics there is the algebraic and matrix formalism). The objective is to arrive at the same result anyway.

This website (and its associated PDF) being still finalized, it necessarily has lacks on convergence controls, on continuity and others... (which will horrify mathematicians ...)! However, I have avoided (or, otherwise, I indicate it) the usual approximations of physics and the use of dimensional analysis, by using it as little as possible. I also try to avoid as much as possible subjects with mathematical tools that have not previously been presented and demonstrated rigorously.

Finally, this presentation, that can still be improved, is not an absolute reference and contains errors. Any comment is welcome. I shall endeavour, as far as possible, to correct the weaknesses and make the necessary changes as soon as possible.

However, while mathematics is accurate and indisputable, theoretical physics (its models), is still interpreted in the common vocabulary (but not in the mathematical vocabulary) and its conclusions all relative. I can only advise, when you browse this website (or associated its PDF), to read by for yourself and not to be subjected to outside influences. You must have a very (very) critical mind, take nothing for granted and question everything without hesitation. In addition, the keyword of good scientist should be: "Doubt, doubt, doubt ... doubt still, and always checks.". We also recall that "nothing that we can see, hear, smell, touch or taste, is what it seems to be", therefore do not rely on your daily experience to draw hasty conclusions, be critical, cartesian, rational and rigorous in your development, reasoning and conclusions!

I want to say to those who would try to find themselves the results of some developments on this website (or its associated PDF), do not worry if they do not success or if they doubt about their competences because of the time spent solving an equation or problem: some theories that seem obvious or easy today, have sometimes needed several weeks, months, even years, to be developed by mathematicians or leading physicists in the past!

I also tried to ensure that this website (and its associated PDF) is pleasing to the eye and to read through. Professional web designers will however excuse me for the poor quality of the HTML / PHP (that they won't see in part ...) / JavaScript / CSS and the abuse usage of the Bevel and Emboss of Photoshop as well for the choise of an interface optimized for a resolution of 1024 x 768 and higher, but I do have enough time to clean up the code and make correct graphic (again, I prefer rather the quality of the content than the container).

Finally, I chose to write this work in the first person plural ("we"). Indeed, the mathematical-physics is not a science that has been made or has evolve through individual work but with intensive collaboration between people connected by the same passion and desire of Knowledge. Thus, by making use of "we", he paid tribute to the dead and missing scientists, to contemporary and future researchers for the work they will perform in order to approach the truth and wisdom.

METHODS

Science is the set of all systematic efforts (scrupulous observations and plausible assumptions until the evidence of the contrary) to acquire knowledge about our environment, to organize and synthesize them into testable laws and theories, whose main purpose is to explain the "how" of things (and NOT the why!) often by a three-step approach:

Scientists have to submit their ideas and results to independent verification and replication of their peers. They must abandon or modify their conclusions when confronted with more complete or different evidences. The credibility of Science is based on this self-correcting mechanism. The history of science shows that this system works very long and very well compared to all the others. In each area, progress has been spectacular. However, the system sometimes failed and has also to be corrected before small drifts accumulate.

The downside is that scientists are humans. They have the imperfections of all humans, and especially, vanity, pride and conceit. Nowadays, it happens that many people working on the same topic for a given time develop a common faith and believe they hold the truth. The leader of the faith is the Pope and distills his opinion. The Pope that plays the game, takes his miter and his pilgrim's staff to evangelize his fellow heretics. Until then, this makes smile. But, as in real religions, they sometimes annoying to want to expand their opinion to those who do not believe. Some of these "churches" do not hesitate to behave like the Inquisition. Those who dare to express a different opinion are burned at every opportunity, during conferences, or at their place of work. Some young researchers, uninspired, prefer to convert to the dominant religion, to become clerics faster rather than innovative researchers or even iconoclasts. The great Pope write his Bible to disseminate his ideas, imposes it to read to students and newcomers. He formats then the thought of younger generations and ensures his throne. This is a medieval attitude that can block progress. Some Popes go so far that they believe be the pope in their specialization field automatically gives them the same throne in all other areas...

This warning, and the reminders that will follow, must serve the scientific to ask himself by making good use of what we consider today as the good working practices (we will discuss the principles of the Descartes method more below) to solve problems or develop theoretical models.

For this purpose, here is a summary table that provides the steps that should be followed by a scientific who works in mathematics or theoretical physics (for definitions, see just below):

Proceed as in the above table is a possible working basis for people working in mathematics and physics. Obviously, proceed cleanly and traditionally as above takes a little more time than doing things no matter how (this is why most teachers do not follow these rules, they don't have enough time to cover the entire course program).

Notes:

R1. Attention, it is very easy to make new physical theories by just aligning words. This is called "philosophy" and the Greeks thought of the atoms in this method. This can lead with a lot of luck to a true theory. Against it is much more difficult to make a "predictive theory", that is to say with equations that predict the outcome of an experiment. Attention, il est très facile de faire des nouvelles théories physiques en alignant des mots.

R2. What separates mathematics and physics is that in mathematics, the hypothesis is always true. Mathematical discourse is not a proof of an external seeking truth, but a target of consistency. What should be correct is just the reasoning.

When these rules are not respected, we speak of "scientific fraud" (which often leads to being fired from his job but unfortunately we still not retired the diplomas when it happens). In general, scientific fraud itself comes in three main forms: plagiarism, fabrication of data and alteration of results unfavorable to the hypothesis, the omission of clear working hypotheses and recolted datas. To these frauds we can also add behaviors that pose problems regarding to the quality of work or more specifically to ethics, such as those aimed at increasing appearance in the production (and through the famous of the scientist) by submitting for example several times the same publication with only a few modificatins, the omission of conflict of interest, the dangerous experiments, the non-conservation of primary data, etc.

DESCARTES' METHOD

Now we present the four principles of the Descartes' method which, as remind, is considered as the first scientific in history by his method of analysis:

P1. Never accept anything as true that I obviously knew her to be such. That is to say, carefully avoid precipitation and to understand nothing more in my judgments than what would appear so clearly and distinctly to my mind, that I had no occasion to doubt.

P2. Divide each of the difficulties I have to examin into as many parts as possible (scrupulous observations and plausible hypothesis until evidence of the opposite), and that would be required to resolve them in the best way.

P3. Driving my thoughts in order, beginning with the simplest objects and easiest to know, to go up gradually by degrees to the knowledge of the most compounds, and even assuming the order between those who not naturally precede each other.

P4. Make everywhere so complete enumerations and so genera reviews, that I'm sure not to omit anything.

OATH OF ARCHIMEDES

Inspired by the Hippocratic Oath, a group of students of the Ecole Polytechnique Fédérale de Lausanne in 1990 developed an oath ofArchimedes expressing the responsibilities and duties of the engineer and technician. It was taken in various versions by other European engineering schools and could serve as basic inspiration oath for scientific researchers (even if there are some important points).

« Considering the life of Archimedes of Syracuse which illustrated as of Antiquity the ambivalent potential of the technique, considering the responsibility increasing for the engineers and scientists with regard to the men and nature, considering the importance of the ethical problems that the technique and its applications raise, today, I pledge following and will endeavour to tend towards the ideal which they represent:

VOCABULARY

Physics and mathematics, like any field of specialization, has its own vocabulary. So that the reader is not lost in the understanding of certain texts he can read on this website (and its associated PDF), we chosed to present here a few fundamentals words, abbreviations and definitions to know.

Thus, the mathematician like to finish his proofs (when he thinks they are correct) by the abbreviation "QED" which means "Quod Erat demonstrandum" (this is Latin).

And during definitions (they are many in math and physics ...) scientist often use the following terminology:

The four are not equivalent (identical in the strict sense). Because "it is sufficient that" correspond to a sufficient condition, but not to a necessary condition.

ON THE SCIENCES

It is important that we define rigorously the different types of sciences which human been often refers. Indeed, it seems that in the 21st century a misnomer is established and that it became impossible for people to distinguish the "intrinsic quality" between a science and another one.

Note: Etymologically, the word "science" comes from the Latin "Scienta" (knowledge) whose root is the verb "scire" which means "to know".

This abuse of language is probably the fact that pure and accurate sciences lose their illusions of universality and objectivity, in the sense that they are self-correcting. This has for effect that some sciences are relegated to the background and try to borrow these methods, principles and origins to create confusion. We must therefore be very careful about the claims of scientificity in the human sciences, and this is also (or especially) true for the dominant trends in economics, sociology and psychology. Quite simply, the issues addressed by the human sciences are extremely complex, poorly reproducible, and empirical arguments supporting their theories are often quite low.

By itself, however, science does not produce absolute truth. By principle, a scientific theory is valid as long as it can predict measurable and reproducible results. But the problems of interpretation of these results are part of natural philosophy.

Given the diversity of phenomena to be studied, over the centuries there has been a growing number of disciplines such as chemistry, biology, thermodynamics, etc. All these disciplines that are a priori heterogeneous have common foundation physics, for language mathematics and for elementary principle the scientific method.

D1. We define as "pure science" any set of knowledge based on rigorous reasoning valid whatever the (arbitrary) elementary factor selected (when we say then "independent of sensible reality") and restricted to the minimum necessary. Only mathematics (often called the "queen of sciences") that can be classified in this category.

D2. We define as "exact science" or "hard science", any set of knowledge based on the study of an observation, observation that has been transcribed in symbolic form (theoretical physics for example). Primarily, the purpose of exact sciences is not to explain the "why" but the "how".

Caution! There is no doubt that the exact sciences have yet an enormous prestige, even among their opponents because of their theoretical and practical success. It is certain that some scientists sometimes abuse of this prestige by showing a sense of superiority that is not necessarily justified. Moreover, it often happens that this same scientists exposed in the popular literature, very speculative ideas as if they were very approved, and extrapolate their results outside the context in which they were tested (and ... under the hypotheses they were checked once...).

Note: The two previous definitions are often included in the definition of "deductive sciences" or even "phenomenological science".

D3. We define as "engineering science" any set of knowledge or practices applied to the needs of human society such as electronics, chemistry, computer science, telecommunications, robotics, aerospace, biotechnology...

D4. We define as "science" any body of knowledge based on studies or observations of events whose interpretation has not yet been transcribed and verified with mathematical rigor, characteristic of previous sciences, but using comparative statistics. We include in this definition: medicine (we should however be careful because some parts of medicine are studying phenomena using mathematical descriptions such as neural networks or other phenomena associated with known physical causes), sociology, psychology, history, biology ...

According to the philosopher Karl Popper, a theory is scientifically acceptable if, as presented, it can be falsifiable, ie subjected to experimental tests. The "scientific knowledge" is then by definition the set of theories that have resisted to falsification. Science is by nature subject to continuous questioning.

D5. We define as "soft science" or "para-science", any set of knowledge or practices that are currently based on non-verifiable facts (not scientifically reproducible) by experience or by mathematics. We include in this definition: astrology, theology, paranormal (which was demolished by science zetetic), graphology...

D6. We define as "phenomenological science" or "natural sciences", any science which is not included in the above definitions (history, sociology, psychology, zoology, biology, ...)

D7. "Scientism" is an ideology that considers experimental science is the only valid mode of knowledge, or, at least, superior to all other forms of interpretion of the world. In this perspective, there is no philosophical, religious or moral truths superior of scientific theories. Only account what is scientifically proven.

D8. The "positivism" is a set of ideas that considers that only the analysis and understanding of facts verified by experience can explain the phenomena of the sensible world. Certainty is provided solely by the scientific experiment. He rejects introspection, intuition and metaphysical approach to explain any knowledge of the phenomena.

What is interesting about this doctrine is that it is certainly one of the few that requires people to have to think for themselves and to understand the environment around them by continually questioning everything and by never accepting anything as granted (...). In addition, the real sciences have this extraordinary property that they give the possibility to understand things beyond what we can see.

But, science is science, and nothing more: a certain ordering, not too bad success, things that no longer leads to the metaphysics as the time of Aristotle, but that does not pretend to give us the whole story on reality or even the bottom of visible things.

TERMINOLOGY

The Table of methods we presented above contains terms that may perhaps seem unknow or barbarians for you. This is why it seems important to provide definitions of these and some other equally important that can avoid important confusion.

D1. Beyond its negative sense, the idea of "problem" refers to the first step of the scientific method. Formulate a problem is also essential for its resolution and allows to properly understand what is the problem and see what needs to be resolved.

The concept of problem is intimately connected to the concept of "assumption" that we will see the definition below.

D2. A "hypothesis" is always, in the context of a theory already established or underlying, a supposition awaiting confirmation or refutation that attempts to explain a group of facts or predict the onset of new facts.

Thus, a hypothesis can be at the origin of a theoretical problem that has to be resolved formally.

D3. The "postulate" in physics corresponds frequently to a principle (see definition below) which admission is required to establish a proof (we mean that this is a non-proovable proposition).

The mathematical equivalent (but in a more rigorous version) of the assumption is the "axiom" for which we will see the definition below.

D4. A "principle" (close parent of "postulate") is a proposal accepted as a basis for reasoning or a general theoretical guide line for reasoning that needs to be performed. In physics, it is also a general law governing a set of phenomena and verified by the accuracy of its consequences.

Note: The word "principle" is used with abuse in small classes or engineering schools by teachers not knowing (which is very rare), or unwilling (rather common), or that can't because lack of time (almost exclusively ) proove a relationship.

The equivalent of the postulate or principle in mathematics is the "axiom" which we define as follows:

D5. An "axiom" is a self-evident proposition or truth by itself which admission is necessary to establish a proof.

Notes:

R1. We could say that this is something we define as the truth for the speech that we argue, like a rule of the game, and that it does not necessarily a universal truth value in the sensitive world around us.

R2. Axioms must always be independent (one should not be able to be prooved from the other) and non-contradictory (sometimes we also say that they must be "consistent").

D6. The "corollary" is a term unfortunately almost nonexistent in physics (wrongly!) and that is in fact a proposal resulting from a truth already demonstrated. We can also say that a corollary is and obvious and necessary consequence of a theorem (or sometimes of a postulate in physics).

D7. A "lemma" is a proposal deduce from one or more assumptions or axioms and that for which the proof prepares this of a theorem.

Note: The concept of "lemma" is also (and this is unfortunate) almost used only in the field of mathematics.

D8. A "conjecture" is a supposition or opinion based on the likelihood of a mathematical result.

Note: Many conjectures have as as little similar to lemmas, as they are checkpoints to obtain significant results.

D9. Beyond its weak conjecture sense, a "theory" or "theorem" is a set articulated around a hypothesis and supported by a set of facts or developments that give it a positive content and make the hypothesis well-founded (or at least plausible in the case of theoretical physics).

D10. A "singularity" is an indeterminacy in a calculation That takes the appearance of a division by zero. This term is both used in mathematics and in physics.

D11. A "proof" is a set of mathematical procedures to follow to prove the result already known or not of a theorem.

D12. If the word "paradox" etymologically means: contrary to common opinion, it is not by pure taste for provocation, but rather for solid reasons. The "sophism" meanwhile, is a deliberately provocative statement, a false proposition based on an apparently valid reasoning. Thus we speak about the "Zeno's paradox" when in reality it is a "sophism". The paradox is not limited to falsity, but implies the coexistence of truth and falsity, so that one can no longer distinguish true and the false. The paradox appears as an unsolvable problem an "aporia".

Note: It should be added that the well-knows paradoxes, by the questions they raised, have permitted significant advances to science and led to major conceptual revolutions in mathematics as in theoretical physics (the paradoxes on sets and on infinity in mathematical, and those at the base of relativity and quantum physics).

SCIENCE AND faitH

We will see that in science, a theory is usually incomplete because it can not fully describe the complexity of the real world. It is thus all theories like the Big Bang (see Astrophysics) or the evolution of species (see Populations Dynamics or Theory of Population or The Games Theory) at least because they are not reproducible under identical conditions.

- "Realism" is a doctrine where physical theories have the aim to describe reality as it is in itself, in its unobservable components.

- "Instrumentalism" is a doctrine where theories are only tools to predict observations but do not describe reality itself.

- "Fictionalism" is the current where the content repository (principles and postulates) of theories is just an illusion, useful only to ensure the linguistic articulation of the fundamental equations.

Even if today the scientific theories are sponsored by many specialists, alternative theories have valid arguments and we can not totally dismiss them. However, the creation of the world in seven days as described in the Bible can no longer be seen as a possible, and many believers recognize that a literal reading is not compatible with the current state of our knowledge and that is more prudent to interpret it as a parable. If science never provides definitive answer, it is no longer possible to ignore it.

Faith (whether religious, superstitious, pseudo-scientific or other) on the contrary is intended to provide absolute truths of a different nature as it is a personal unverifiable belief. In fact, one of the functions of religion is to give meaning to the phenomena that can not be explained rationally. Progress of knowledge trough science therefore cause sometimes questioning the religious dogma.

A contrario, sauf à prétendre imposer sa foi (qui n'est autre qu'une conviction intimement personnelle et subjective) aux autres, il faut se défier de la tentation naturelle de qualifier de fait scientifiquement prouvé les extrapolations des modèles scientifiques au-delà de leur champ d'application.

The word "science" is, as we have already mentioned above, increasingly used to argue that there is a scientific evidence where there is only a belief (some web pages like this proliferate always more and more). According to its detractors it is, for example, the case of the movement of Scientology (but thera are many others). According to them, we should rather speak about "occult sciences".

The occult sciences and traditional sciences exist since antiquity; they consist on a series of mysterious knowledge and practices designed to penetrate and dominate the secrets of nature. Over the past centuries, they have been progressively excluded from science. The philosopher Karl Popper has longly questioned himself about the nature of the demarcation between science and pseudoscience. After noticing that it is possible to find observations to confirm almost any theory, he proposes a methodology based on falsifiability. A theory must according to him, to deserve the adjective "scientific", guarantee the impossibility of some events. It becomes therefore rebuttable, so (and only then) capable of integrating science. It would suffice to observe any of these events to invalidate the theory, and therefore take the way to improving it.

Finally, we would like to quote Lavoisier: "The physicist may also, in the silence of his laboratory and his cabinet, perform patriotic functions; he can thanks to his works reduce the mass of evils which afflict happiness and, had he not, contributed by the new roads that he opened to himself, only to delay of a few years, of a few days, the average life of humans, he could also aspire to the glorious title of benefactor of humanity. "