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We know nothing, not even whether we know or do not know, or what it is to know or not to know, or in general whether anything exists or not, Metrodorus of Chios

Even before Plato Greek philosophers tried to find the limitations of our knowledge. Is that what we see the reality? Why is this so interesting and why should we consider this problem? Maybe an answer is the story of Charly in "Flowers for Algernon". In his progress notes Charly writes: Before the operation I was looking the world through a keyhole inside a dark room. My opinion was that I can understand the world by trying to read some books that I could not understand. Maybe if I try harder I can understand what is written.. One of the books has a strange title: "The lost paradise".....

The prisoner's in Plato's cave are the poor ordinary men, like me and you. Fortunately there are philosophers who are not happy with what they see or hear and they turn their heads and even try to get out from the cave to see the "real world" Today the situation is even worse, there are some who control the fire, the projector of the images and finally what we see in order to control us (at least they try).

In the seventh book of "The Republic" the Greek philosopher Plato discusses our situation of the understanding of the world. In an allegoric view we live in a cave having our legs and necks chained so that we cannot turn around our heads and can see only before us. Above and behind us a fire is blazing at a distance, and between the fire and us there is a raised way; and we see a low wall built along the way, like the screen which marionette players have in front of them, over which they show their puppets. The situation is like in a movie theatre where we observe the shadow of objects on a wall using as a projector the light of a blazing fire. From these limitations we try our best to understand the world from the shadows of the objects.

Socrates: And now, let me show in a parable how far our nature is enlightened or unenlightened: Behold! human beings living in an underground den, which has a mouth open towards the light; here these people have been from their childhood, and have their legs and necks chained so that they cannot move, and can only see before them, being prevented by the chains from turning their heads. Above and behind them a fire is blazing at a distance, and between the fire and the prisoners there is a raised walk; and you will see, if you look a low wall built along the walk, like the screen which marionette players have in front of them, over which they show the puppets.

Glaucon: I see. Socrates: And do you see men passing along the wall carrying all sorts of containers, and statues and figures of animals made of wood and stone and various materials, which appear over the wall? Some of them are talking, others silent.

Glaucon: You have shown me a strange image, and these are strange prisoners.

Socrates: Like ourselves. And they see only their own shadows, or the shadows of one another, which the fire throws on the opposite wall of the cave.

Glaucon: True; how could they see anything but the shadows if they were never allowed to move their heads?

Socrates: And the prisoners, would they not see only the shadows of the objects which are being carried?

Glaucon: Yes.

Socrates: And if the prisoners were able to talk with one another, would they not suppose that they were naming what was actually before them?

Glaucon: Very true.

Socrates: And suppose further that the prison had an echo which came from the other side, would the prisoners not be sure to fancy when one of the passers-by spoke that the voice which they heard came from the passing shadow?

Glaucon: No question.

Socrates: To them the truth would be literally nothing but the shadows of the images.

Glaucon: That is certain.

Socrates: And now look again, and see what will naturally follow if the prisoners are released and disabused of their error. At first, when any of them is liberated and compelled suddenly to stand up and turn his neck around and walk and look towards the light, he will suffer sharp pains; the glare will distress him, and he will be unable to see the realities of which in his former state he had seen the shadows; and then conceive someone saying to him, that what he saw before was an illusion, but that now, when he is approaching nearer to being and his eye is turned towards more real existence, he has a clearer vision—what will be his reply?

And you may further imagine that his instructor is pointing to the objects as they pass and requiring him to name them—will he not be perplexed? Will he not fancy that the shadows which he formerly saw are truer than the objects which are now shown to him?

Glaucon: Far truer.

Socrates: And if he is compelled to look straight at the light, will he not have a pain in his eyes which will make him turn away to take refuge in the objects of vision which he can see, and which he will conceive to be in reality clearer than the things which are now being shown to him?

Glaucon: True.

Socrates: And suppose once more, that he is reluctantly dragged up a steep and rugged ascent, and held fast until he is forced into the presence of the sun itself, is he not likely to be pained and irritated? When he approaches the light his eyes will be dazzled, and he will not be able to see anything at all of what are now called realities.

Glaucon: Not all in a moment.

Socrates: He will require to grow accustomed to the sight of the upper world. And first he will see the shadows best, next the reflections of men and other objects in the water, and then the objects themselves; then he will gaze upon the light of the moon and the stars and the spangled heaven; and he will see the sky and the stars by night better than the sun or the light of the sun by day.

Glaucon: Certainly.

Socrates: Last of all, he will be able to see the sun, and not mere reflections of it in the water, but he will see the sun in its own proper place, and not in another; and he will contemplate it as it is.

Glaucon: Certainly.

Socrates: He will then proceed to argue that it is the sun who gives the season and the years, and is the guardian of all that is in the visible world, and in a certain way the cause of all things which he and his fellows have been accustomed to behold?

Glaucon: Clearly, he would first see the sun and then reason about it.

Socrates: And when he remembered his old habitation, and the wisdom of the den and his fellow-prisoners, do you not suppose that he would congratulate himself on his improvement, and pity them?

Glaucon: Certainly, he would.

Socrates: And if the prisoners were in the habit of conferring honors among themselves on those who were quickest to observe the passing shadows and to remark which of them went before, and which followed after, and which were together; and who were therefore best able to draw conclusions as to the future, do you think that he would care for such honors and glories, or envy the possessors of them? Would he not say with Homer, "Better to be the poor servant of a poor master," and to endure anything, rather than think as they do and live after their manner?

Glaucon: Yes, I think that he would rather suffer anything than entertain these false notions and live in this miserable manner.

Socrates: Imagine once more such a one coming suddenly out of the sun to be replaced in his old situation; would he not be certain to have his eyes full of darkness?

Glaucon: To be sure.

Socrates: And if there were a contest, and he had to compete in measuring the shadows with the prisoners who had never moved out of the den, while his sight was still weak, and before his eyes had become steady (and the time which would be needed to acquire this new habit of sight might be very considerable), would he not be ridiculous? Men would say of him that up he went and down he came without his eyes; and that it was better not even to think of ascending; and if anyone tried to free another prisoner and lead him up to the light, let them only catch the offender, and they would put him to death.

Glaucon: No question.

Socrates: If I am right, certain professors of education must be wrong when they say that they can put a knowledge into the soul which was not there before, like sight into blind eyes.

Glaucon: They undoubtedly say this.

Socrates: Whereas, our argument shows that the power and capacity of learning exists in the soul already; and that just as the eye was unable to turn from darkness to light without the whole body, so too the instrument of knowledge can only by the movement of the whole soul be turned from the world of becoming into that of being, and learn by degrees to endure the sight of being, and of the brightest and best of being, or in other words, of the good.

Glaucon: Very true.

Socrates: Each of you, when his turn comes, must go down to the general underground abode, and get the habit of seeing in the dark. When you have acquired the habit, you will see ten thousand times better than the inhabitants of the den, and you will know what the several images are, and what they represent, because you have seen the beautiful and just and good in their truth. And thus our State which is also yours will be a reality, and not a dream only, and will be administered in a spirit unlike that of other States, in which men fight with one another about shadows only and are distracted in the struggle for power, which in their eyes is a great good. Whereas the truth is that the State in which the rulers are most reluctant to govern is always the best and most quietly governed, and the State in which they are most eager, the worst....

When a person starts on the discovery of the absolute by the light of reason only, and without any assistance of sense, and perseveres until by pure intelligence he arrives at the perception of the absolute good, he at last finds himself at the end of the intellectual world, as in the case of sight at the end of the visible....

Because a freeman ought not be a slave in the acquisition of knowledge of any kind, bodily exercise, when compulsory, does no harm to the body; but knowledge which is acquired under compulsion obtains no hold on the mind.

Glaucon: Very true.

Socrates: Then, my good friend, do not use compulsion, but let early education be a sort of amusement; you will then be better able to find out the natural bent.

The enlightenment process, “The divide line Schema” of Plato. We should not forget the return to the cave at the end, to tell the others in the cave what the reality is even if the danger is that they, as Plato probably used an allegory for Socrates, not only refuse to believe you but even kill you.

Today we consider Plato's problem as an inverse problem to solve. Such a situation is common as we cannot see inside the objects directly but have to find their structure from measurements outside the objects. These observations often have errors like the fire in the cave which produces a projected distorted blurred image similar to the light of stars that is distorted passing through the atmosphere. The term inverse problem is a well known mathematical term dating at least from the 19th century and there are several mathematical journals devoted to this topic. What is meant in simplistic terms: first you know the ideal answer, and second you take into account any constraints and mathematically determine the optimum parameter values to provide the ideal answer. In other words you have the result and the inverse problem is to determine the cause of this result. We have to solve inverse problems in geophysical studies and medical applications where it is often desired to find out the internal unknown structure of an object using measurements performed outside the object, that is, non-non-invasive. This can be done e.g. by measuring radiation, X-ray, ultrasound etc., or acoustic waves that passes through the object. The object under study can be e.g. the human body, or the earth crust.

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