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Thinking is not the only way to learn to think: keys to scientific thinking

Author: Héctor L. Mancini, Universidad de Navarra
Published in: conference proceedings de las IX conference de Innovación Pedagógica de Attendis. Mathematics for life in a digital world. Granada: Attendis, 2009. 205-20. Closing conference. IX conference of Pedagogical Innovation. Granada, 14 March 2009
Date of publication: 2009

Human societies have always attributed great value to knowledge and wisdom. The knowledge, a product of the human mind, originates in the senses and their extensions and continues abstractly through logical reasoning and reflection.

In all cultures, a great deal of knowledge is acquired, preserved and transmitted, knowledge which at each historical moment is considered vital for survival, and also other knowledge which, without being strictly necessary for life, contributes to enriching the human spirit. These apparently superfluous activities, such as art, abstract thought or recreational activities, although they do not respond to direct needs, are inherent to human beings and contribute to their spiritual growth. *(1).

Among them we highlight scientific thought, when viewed in its most basic aspects, as an activity whose purpose does not obey a specific application, i.e. as a cognitive activity. Thus considered, scientific thought has similarities and differences with other rational activities such as philosophy, theology, sociology, psychology, history and, in general, with all those disciplines that we call "Humanities". The study of reality, when explored in all its depth, requires all these perspectives, each with its own methodology.

In order to analyse what aspects are particular to scientific thinking, we first need to define what the object of its study is and how it does so. We can ask this question in a very general way if we ask ourselves about the subject of studies that are carried out in a science School of any university. A simple and concrete answer would tell us that the behaviour of subject is studied there, with a particular method known as the method of experimental sciences.*(2)

A science is experimental*(3) when, on the one hand, it takes data from reality (or from an experiment that represents it), and analyses what happens after the action of one or more causes. With the results, he elaborates a mathematical theory or model, which explains how that cause acting on that system produces certain effects. This theory can then be corrected based on new experimental data, and repeating the procedure until the numbers obtained with the model and those measured in the experiment coincide. There will always be an error, which will be smaller and smaller. The error or uncertainty with which this behaviour is known is determined by the Degree of coincidence between both results.

We can already see from this brief summary, that there are some important keys to follow, which characterise the method. Firstly, we need two ways or phases of approaching the object under study that we call "system", in the manner of classical thermodynamics (for thermodynamics, a system is any portion of the universe that we assume to be isolated for study).

In the first phase, we create a mathematical model that should provide the observed effects from testable causes. In the second, we proceed to experimental verification by measuring the relevant data for the theoretical model , either from reality itself, or in a specially designed experiment to observe whether the action of these causes always produces the same effects. The results of these measurements must be expressed mathematically.

When the theoretical and experimental results coincide, we can safely say that the subject under study always responds in the same way, and that the mathematical model represents this behaviour.

With this procedure we convert natural phenomena or natural entities into entities of reason, into ideal concepts. From there, we can proceed with all the rules of mathematics and make predictions by changing the values that are modifiable. If the coincidence holds, we say that we are in possession of a certain knowledge , of a truth about nature that we call law. A truth that we know with a certain Degree of certainty. The laws obtained with this procedure, are then integrated with others, in ever wider circles, to obtain a whole theory about some aspect of reality.

This is the essence of the scientific approach. Mathematical models, once validated, make it possible to justify the past, predict future behaviour or analyse what would happen if some parameter were to change.

If we observe carefully, according to this method truth always resides in things, while certainty resides in the knowing subject. It is possible to know with great certainty something false or, conversely, to be uncertain about a result that is true. Truth is pragmatic, and we can be sure that the knowledge is true, even if we are not very certain about it. This means that nature always behaves in a certain way, although we can make mistakes in verifying it.

When these results are verified by many observers, we also say that we have a knowledge goal (actually, it would be inter-subjective, but we will not enter into philosophical discussions that may divert us from our purpose).

We will conclude this introduction by saying that experimental science is both analytical and deductive, i.e. it goes indistinctly from effects to causes and from causes to effects, without any hierarchy in the procedures. There are only physicists who are more specialised in one or the other aspect ("theoretical" or "experimental") and the important thing is the final coincidence in the results.

Mathematics and experimental science

In direct relation to this congress, we stress that for experimental science, mathematics is a language. Mathematics is not an experimental science, nor is it an end in itself, a statement that when made in this context becomes a mistake, and it is a very common mistake that we must avoid.

Nor can it be said that "nature is written in mathematical language" (like G. Galilei). Nature is mute, it is not written in any language and it is man who expresses it. But the role of mathematics as the language of science is so important that we can say without doubt that without mathematics experimental science would be impossible. Later on we will see some clarifying examples.

It is necessary to distinguish between this conception of mathematics as a language and that of mathematics as a science in its own right*(4). In the latter context, mathematics is a science that deals with relating abstract concepts to each other, without reference to reality. That is to say, ideal entities that do not necessarily correspond to real things. In this case, the logic of speech is the appropriate and necessary instrument for assessing whether or not the reasoning is correct. A subject of logical "truth" that has no relation to the "adequacy" between ideas and the real world.

These few reflections suffice to get to the heart of the problem, which we focus on the following three questions:

a. What is thinking important for?

b. What characteristics of scientific thinking are useful for any rational perspective?

c. How to incorporate the habit of thinking that characterises science into other teaching disciplines at different levels?

a. What is thinking important for?

The importance of thought is transcendent. We humans are biologically classified as homo sapiens, and on a purely biological level it is the use of reason coupled with free will that distinguishes us from animals. We live in the world and seek to understand it.

We usually emphasise this distinction philosophically, saying that the human being is a person, a being endowed with intelligence and freedom. Intelligence is a way of knowing and freedom is a way of acting, which should be guided by this knowledge. From this we can immediately conclude that if society is interested in responsible social behaviour, it must increase the quality of its thinking. The use of intelligence is the basis for free action and, unfortunately, there are no known methods to increase it. But man has known for millennia that true human progress is based on high quality education, i.e. education for better thinking.

These simple reasons justify the concern that human societies (even outside politics) have always shown for finding the most effective method of transmitting thought from one generation to the next. The question of knowledge, thinking and its teaching is therefore a problem whose solution has an enormous influence on society as a whole. Control over the dissemination of certain knowledge can become an instrument of power, and many social groups (indeed whole countries) have exercised this control fiercely. Dictatorships of every subject and colour that humanity has known are a sad testimony to this.

That is why teaching how to think, if it is really possible, would probably be the most important way of teaching how to be free. Human beings, with the power of their thoughts, are able to feel free even when they are slaves, and conversely, they can feel like slaves even when they are free. How is it possible that something as abstract as thought can have such an influence on bodies Materials made up of 80% water molecules? How is it possible for these beings, in objective and palpable circumstances of oppression, to feel free as a result of ideas that are basically nothing more than words?

We must accept that words have power, that they are creative and that they tend to produce what they mean. Human thought makes this and much more possible, and we will try to demonstrate this through some examples.

But first we must distinguish between ideas and information. In education, any subject has a structure of original ideas, which are usually few, consequences of those ideas, which are usually many, and a broad and vast content of information (data).

Today, one can get a good idea of the amount of information produced by humans, for example, by measuring data traffic on the internet. The results clearly show an exponential growth, reflected in the permanent demand for ever increasing bandwidths. This simple verification is enough for us to understand that accumulating data and teaching how to remember it is an almost useless task for the human mind, which is permanently overtaken by computers. The centre of gravity of teaching has shifted from the accumulation of data to the selection, classification, processing and increased speed of access to them. More important than the quantity of information is knowing where it is and how to handle it, and this leads us into the world of ideas. We need to ask ourselves what we want that information for and reflect on its use. We need to stop and think.

Despite the significant growth in the amount of information available, teachers with some years of experience have not noticed any qualitative improvement in the basic aspects of thinking, i.e. in the generation of ideas. Rather, we have found that more information does not imply an improvement in quality. The exponential increase in communications and its comparison with everyday behaviour shows us that increasing information without improving thinking is not enough; mere technological progress is not enough financial aid to think better. What is more, technology sometimes acts inversely, producing backwardness.

We have all surely noticed the impoverishment of language brought about by the use of mobile phones to send messages. The poor understanding of texts and the expression of ideas that we observe today in all areas has one of its causes in this mutation of language. The deterioration of language diminishes the ability to think.

Critical and attentive reading and the practice of written expression are fundamental pillars for the construction of thinking and for improving teaching and learning. In them lies one of the keys that can help build quality thinking.

We said earlier that improving thinking financial aid to be free. In the light of the circumstances we have just mentioned, we can affirm that educators who, in addition to presenting the concrete contents with the information of their subject, teach how to process it and to reason from this data, become agents for the construction of a better and freer society. Once we learn to think for ourselves, there is no limit to our thinking other than our own capacity. Therefore, if we teach our students to think for themselves on the basis of the contents of our subject, we will be teaching them to live and act agreement to an end chosen by reason, to live reasonably, which is to say, responsibly.

This task has its award. We will simultaneously find that with the personal growth of our students to which we will have contributed, the foundation of our authority as teachers will appear. Let us remember that the Latin "autoritas" has its Greek root in "augere", i.e. to help to grow. Fulfilling this purpose, helping to grow, is therefore the basis of authority.

Therefore, the task of teaching how to think in the classroom, should be a priority occupation as opposed to the mere accumulation of data, to recipes for practical use, or to semi-elaborated thoughts that only report a result without detailing the path by which it has been obtained, or that do not analyse its Degree of difficulty and the scope of validity or verifiability.

Contemporary society needs to encourage the exercise of reflective thinking from the earliest grades. In today's society, the teaching of rational ways of thinking has become a matter of survival and it is in this context, that of reflective thinking, that the methods of scientific thinking can collaborate most effectively.

b. Some keys to scientific thinking

What characteristics of scientific thinking are useful to any rational perspective?

We will try to answer this question by means of some examples. The historical evolution of some ideas related to what we have called "keys to scientific thought" sample patterns that could be useful in other fields. We will place these examples in three scenarios, corresponding to each of the worldviews*(5) into which the history of human thought is usually grouped in a simplified way. Within each paradigm*(6) we will try to show the role played by mathematics within experimental science.

The first example, the solution of the problem of the curvature of the Earth, which took place within the so-called "organicist" worldview of Aristotelian philosophy,is intended to highlight the limits of scientific thought*(7).

A reflection on the force of gravity, which we will consider secondly, will allow us to appreciate the value of the axioms we postulated before starting the thought process. This problem was detected in antiquity, but was not solved until development of the deterministic paradigm initiated by Newton and culminated by Laplace.

Finally, situated in the transition from determinism to contemporary evolutionary (or naturalistic) thinking, we will try to show the creative value of mathematics as a language by describing a problem that was solved theoretically by A. Einstein 50 years before its first practical application.

First example: The Earth's surface must necessarily be curved.

In order to start thinking, data are needed and it is worth commenting a little on how to obtain them and their role in reflective thinking. The data necessary to begin to think about questions that later become basic for knowledge, usually arise more from an almost childish curiosity that leads to transforming facts that are inconsequential for many into a problem for a few, rather than from a need.

Eratosthenes*(8), the famous Greek thinker of the 3rd century BC, was struck by a piece of information that was passed on to him. A cane which cast a shadow at midday in his city, Alexandria, did not cast a shadow that same day in Siena (now Aswan), Egypt. It is a very famous anecdote.

This fact was enough for Eratosthenes to conclude that, if the fact was true, the earth could not be flat. He sent out to verify the fact and began to think. Using simple proportions between known objects: the shadow and height of an obelisk in Alexandria, the distance between cities and the radius of the earth (the unknown), he concluded that if the earth were a perfect sphere, then it should have a radius of about 6400 km, i.e. with a circumference at the equator of about 40,200 km in perimeter.

The result is not bad at all, despite the scarce technology available at that time to verify these results by other methods. agreement In today's astronomical tables the Earth appears with a radius of 6370 km, and although historians do not quite agree on the exact measurement calculated by Eratosthenes, the problem could be considered brilliantly solved.

Fifty years after his death, Eratosthenes was joined by a perfectionist disciple called Poseidonius, who re-measured and re-calculated according to the same procedure and obtained as result 28,800 km for the perimeter. A very different value from that of his master. To the experts of the time this figure was more familiar than that of Eratosthenes, but they had no other evidence and according to the known land and sea areas a smaller planet was more "credible", so they accepted this value. Between 127 and 151 AD, Claudius Ptolemy, who synthesised the knowledge of his time, recorded the latter figure as true and from there it was passed on to all known schools of astronomy.

The story continues many centuries later with Paolo ToscaneIli, a Florentine physician and draughtsman, born in 1445. Thanks to his skill for the graphic design he became a cartographer and went down in the history of science like many others, thanks to a huge mistake he made. Toscanelli drew some maps with the scales of the school of Alexandria, which assigned to the circumference of the Earth almost a third less than the value calculated by Eratosthenes and added his errors. With Toscanelli's distance, the eastward journey to the west seemed not only feasible, but also relatively easy since the Asian coast was situated about 5000 km from Europe and islands close to each coast were known (Canary Islands, Madeira and the Azores and Japan (Cipango) on the Asian side).

Christopher Columbus*(9) was able to see these maps and that was enough, his spirit of adventure did the rest. Travelling west by sea avoided the difficulties of the "spice route", which required crossing the territories occupied by the Turks, or sailing along the coasts of Africa, which were guarded by the Portuguese.

Columbus thought it was a brilliant business and went in search of a capitalist partner , an investor. He began by trying to convince the King of Spain, who left the problem in the hands of a commission of experts. The experts showed that Columbus was making several mistakes (in addition to those on Toscanelli's map). He confused the Arabian mile with the Italian mile, which were different, thus placing the Asian coast even closer.

In the face of the refusal, Columbus was not discouraged. He went to try to convince the King of Portugal who immediately, as was his custom, appointed another commission of wise experts with the result that we know. Once again, they proved Columbus' mistakes and again there was no funding.

Wrong or not, Columbus was convinced of his business and turned again to the King of Spain and obtained an interview in San Fernando, near Granada. And as we all know, the commission and the crown decided to support him. Columbus was able to travel, set foot on land and died believing he had reached the shores of Asia. A fascinating and adventurous story built on a gross miscalculation.

If we consider history from our framework, the problem of the curvature of the Earth was correctly solved (more than acceptably) two centuries before Christ by Eratosthenes. But a series of beliefs based on other unreliable data led to a solution with a larger error being accepted as valid. A millennium and a half later the error had increased until the solution was completely incorrect. The academies knew it. But one misguided man, making error after error but endowed with a tenacious will, persisted until he secured funding for a business, which by all accounts seemed irrational and necessarily destined to fail.

But in the wrongly calculated position on the coast of Asia, there was a new reality that no one knew about.

Today we remember this story, as the theoretical error that led to one of the greatest successes in human history, the finding of America, and we also learn that scientific thought never exhausts reality (and much less life).

Second example: Things fall down...

If a scientific description of the Universe is attempted, the first problem to be solved is that of motion. Once we know some data, for example, the distance to be travelled, we must be able to establish what additional data would be needed to predict some result that interests us, such as the time it would take to travel that path, or the position in which we would find ourselves after a certain amount of time. This involves establishing quantitative relationships (equations) between the variables that describe it. But before we start this work, we need to know even more.

If we imagine that we are in a boat floating in the middle of a calm lake and we see another boat approaching without anyone pushing it, we cannot know without an external reference which of the boats is moving. Without such a reference we cannot say absolutely whether one or both are moving. We must be content to say that they are moving relatively, i.e. one relative to the other. This means that in order to express the motion we first need a reference system.

If it were possible to imagine the existence of an absolute, immobile reference system, then we could make statements like "the earth moves" and imagine the earth is rotating about itself, within an empty, unchanging space. If we have no such reference, we can only assert that it moves relative to something else.

That subject of immutable space where we can imagine that bodies are found Materials, is an empty space invented by Euclid*(10) and is very intuitive for us today. We say that a space is isotropic and homogeneous when its geometric properties do not change in any direction (isotropic) nor do they change from one point to another when we travel through it in any direction (homogeneous). In this space, time and spatial coordinates are separable and independent. We can say with Aristotle that "time is the measure of motion".

To describe the motion in that space, it would be sufficient to consider any point as a reference and measure from there the position of the mobile whose motion we wish to analyse, and then relate it to the causes that produce it. Without considering any details, we can see that if space has these properties, then, all things being equal, the motion should be independent of the direction in which it is produced.

And this is what happens in our everyday reality when we consider movement forwards or backwards, to the left or to the right. But for some reason, that doesn't happen when the body moves up or down. The things we let free in this space always fall downwards. Never forward or backward, never left or right, they fall down.

Therefore, in our world there is a particular direction which is not the same as the others, and unless we can explain why this is so, we cannot use Euclidean space, which we proposed earlier to describe motion.

Aristotle, who was a very intelligent man, when he compiled all the physics known in his time, recognised this fact and thought, with many others, that if it was necessary to place a reference system, the most suitable place was that point towards which all things naturally fall, that is, at the centre of the earth. Thus geocentrism entered history.

Logically and within that reasoning, the centre of the Earth should also be considered as the centre of the planetary system (this is the famous Ptolemaic system). Already in antiquity there were other alternative proposals such as that of Aristarchus of Samos, who proposed a system with the Sun at the centre (heliocentric), but the description of the motions of the planets known at that time was too complicated, in particular the motion of the Moon, our nearest star and whose motion was necessary to know very well, as it was the basis of many calendars. The trajectories of the other planets looked very strange when viewed from the earth, but they were subject of specialists, not of calendars.

For these reasons the heliocentric system lay dormant until N. Copernicus refloated it and Johannes Keppler demonstrated its usefulness in describing planetary motions. Then Galileo Galilei, with his newly invented telescope, discovered that the moon was not the only satellite revolving around a planet.

As we know, a great controversy arose with the Copernicanists on the one hand and the advocates of the geocentric system on the other. This system became identified with the person of Aristotle for reasons that in reality had little to do with science. Aristotle, as is well known, was "baptised" 1500 years after his death by St. Thomas Aquinas, who used his philosophical system within theology. This theology was later called "scholasticism", i.e. "that which was learned in school" and the schools at that time were almost exclusively promoted by the Catholic Church. This fact placed Aristotle at the centre of an argument between religionists and atheists. In the centuries after Newton, the "backwardness" of science up to then was permanently attributed to "his authority". In plenary session of the Executive Council 20th century, and despite the fact that the gravitational system of Classical Mechanics had already been absorbed by A. Einstein's General Relativity, many literati still continued with this idea*(11).

One wonders whether, during those long and (supposedly) dark centuries before Newton, there was no one intelligent and courageous enough to correct that mistake (nowadays, in science, such mistakes are usually corrected within a few years). And in my opinion, believe it or not, there was not.

There was no objection because the objection was and is very serious, and moreover, because it derives from a fact that is very easy to verify. With or without a background in physics, anyone can see that things move in one way in a horizontal plane, but on the vertical axis... they always fall downwards!

The importance of this problem was not fully understood until the work of Isaac Newton, who for the first time correctly described the problem of motion while at the same time explaining why things fall downwards on the Earth. That is to say, alongside the laws of motion, Newton posed a special law for the vertical axis: the law of "gravity".

In Newton's Laws (motion and gravity) there is a property of subject called "mass". Newton realised that if the "mass" used in the laws of motion on Earth was the same property that he should use to calculate the force of gravity on any planet, then he could also explain why, near the surface of the Earth, all bodies, large or small, always fall with the same acceleration. A very simple calculation (which today can be seen in all textbooks), explaining a fact that had been carefully verified in Galileo's experiments many years earlier.

From then on, by means of Newton's laws, it was possible to describe the motion on Earth and in "the heavens" with one and the same theory. Two realities that were considered to be different at the time.

This result signified the birth of a new scientific paradigm, determinism, which was projected into all fields of thought, including philosophy. Years later, Laplace expressed the predictive power of these calculations by saying that, if at a given instant, an observer could know the positions and velocities of all the particles in the Universe, he would simultaneously know its entire history and its entire future.

Laplace was too intelligent to imagine such a thing within human possibilities knowledge. That is why this hypothetical omniscient observer is often called "Laplace's little devil"*(12).

We want to emphasise with this example, that from an adequate and rigorously controlled axiomatics (such as that provided in classical physics by Newton's laws) the quality of thought and the quantity of correct conclusions that we can obtain increase. It is therefore very important for our purpose to put emphasis on carefully selecting and controlling the axioms that we will use in the construction of our thinking.

Third example: How much is an idea worth?

At the beginning of the 20th century, certain problems appeared in physics that would lead to a new scientific paradigm. The theoretical landscape constituted by deterministic mechanics, together with the great synthesis of electromagnetism by J. C. Maxwell and the development of "Equilibrium Thermodynamics", which had been elaborated during the whole of the 19th century, seemed to explain everything. However, new evidence showed that the edifice was beginning to fail on several fronts.

One of the key problems studied at that time was the so-called "Black Body Radiation"*(13). Classical electromagnetism led to an erroneous conclusion, which came to be dramatised as "the ultraviolet catastrophe". Although the mathematical details are beyond the scope of this lecture, the underlying ideas are very simple and we will try to recount them without using any formula.

It is common experience that if we put a piece of iron on the fire, it tends to reach the colour of flame. It starts to turn brown, then red, and if we increase the fire too much, it turns whitish. If we remove it, we can see that it emits light with a colouring in accordance with the temperature it has reached. This fact, known since ancient times, had been correctly interpreted by electromagnetism, saying that light was an electromagnetic wave visible to the human eye and composed of different colours. Newton, using a prism, had shown that the white light from the sun was broken down into these colours, the same colours that make up the rainbow.

When we look at hot iron, we see that the colour composition is due to the temperature of the body. The set of colours obtained is called the electromagnetic spectrum, of which the human eye is only able to observe a small part, the part between red and violet. Radiations below red are called infrared and those beyond violet are called ultraviolet.

Classical electromagnetic theory gave an explanation for the distribution of colours with temperature with good agreement for frequencies near the red and infrared region, but completely departed from reality in the ultraviolet region (hence the title "the catastrophe...").

By means of a bold hypothesis (that energy can only be exchanged discretely, i.e. at values that are an integer multiple of a certain minimum quantity), Max Planck succeeded in 1901 in explaining the correct shape of the curve that gives the intensities of the colours that appear at a certain temperature. Planck's success gave birth to a new mechanics: Quantum Mechanics, which revolutionised the whole of 20th century physics. Thanks to quantum mechanics, the atomic and nuclear world could be understood in great detail and its consequences contributed to changing the classical deterministic paradigm.

For our history, it is enough to know that a decade after Planck's work, Albert Einstein wanted to deduce this formula again, but this time, from the basic processes that could occur in an atom when it exchanges energy with radiation.

Until then, only two processes were known: absorption and spontaneous emission. In the presence of a radiation field, an atom could only absorb energy if its value coincided with the value of that fundamental energy and then remain for a certain time in an excited state with that excess energy.

Then, spontaneously, the atom returned to equilibrium by returning the excess energy it had accumulated to the environment (today we say by emitting a photon).

Einstein could not reproduce Planck's formula with these two processes alone. He realised mathematically, that to reproduce the formula, he needed to add one more term. Every term in an equation is always identified with some physical reality, and nothing indicated that there was any other process in reality. However, A. Einstein, trusting in the coherence of mathematics, added that term to the equation and with this addition, Planck's formula could be demonstrated in an exact and simple way.

Consequently, this aggregate term must correspond to a phenomenon that had never been observed before. Einstein was confident of its existence and named it stimulated emission.

This would be just a nice academic exercise if it were not for the fact that 20 years later, after much effort and research, it was possible to verify that this phenomenon existed. And then, in 1960, the first practical device based on "stimulated emission" was built: the laser.

The word LASER, an acronym for Light Amplification by eStimulated Emission of Radiation, expresses the origin of this device, whose original idea was due to a genius physicist's respect for mathematical coherence.

Today in every home in the developed world there is a laser in a computer, a CD player, or a DVD player. This ingenuity is the basis of an industry now worth more than twenty billion dollars a year.

How much was that idea worth 80 years ago...?

c. Conclusion

Can scientific thinking contribute to the teaching of any subject?

We have presented several cases that illustrate the development of ideas that we believe are sufficient to illustrate the power and precision of the scientific method. We hope that with the above considerations and examples it has become clear to us that:

  • Scientific thinking does not explain everything, but it follows a methodology that makes it very reliable and has some keys that can be very useful in the teaching of any subject subject.

  • In our opinion, the basics to be transmitted in teaching are:

    • Learning to observe attentively.

    • To express ideas precisely.

    • To seek explanations (theories) by tracing back to causes.

    • To verify theories through direct or indirect evidence.

    • To put theories on test analysing consequences and looking for alternatives.

  • The most important goal should always be to awaken our students' enthusiasm for knowledge.


  1. B. Russell "The Wisdom of the West", 2nd Edic Aguilar, Buenos Aires (1964).

  2. M. Artigas "Filosofía de la Ciencia Experimental", EUNSA, Pamplona, 2nd Ed. (1992).

  3. M. Bunge "Science, its Method and its Philosophy", Metascientific Queries (Springfield, Ill. Charles C. Thomas, 1959). One of four essays published separately in Spanish (University of Buenos Aires, 1958).

  4. L. Santaló "La Matemática, una Filosofía y una Técnica", Ed. Ariel, Barcelona (1994).

  5. We use the word cosmovision in the sense given by M. Artigas in: "La Mente del Universo", EUNSA (1999). A less restrictive sense, although similar to T. S. Kuhn's word "paradigm".

  6. T. S. Kuhn, "The Structure of Scientific Revolutions". Fondo de Cultura Económica, Mexico (1971).

  7. M. Artigas, "El desafío de la Racionalidad", 2nd Ed. EUNSA, Pamplona (1992).

  8. I. Asimov, "Enciclopedia Biográfica de Ciencia y Tecnología". Revista de Occidente, Madrid (1973).

  9. I. Asimov, "Enciclopedia Biográfica de Ciencia y Tecnología". Revista de Occidente, Madrid (1973).

  10. I. Asimov. Work cited.

  11. For example, E. Sábato, (Argentinian writer) wrote in "One and the Universe" (1945), as a definition, "Telescope: apparatus used to observe distant objects and to refute Aristotle" ...

  12. Both "Laplace's imp" and "Buridan's ass" do not allude to their authors (ggg).

  13. Grant R. Fowles, "Introduction to Modern Optics", pp. 204, 2nd. Dover Publications, N. York 1989.