The Discovery of the School of Athens Part 3: The Geometric Proof

By Gerald Therrien

Refer here for Part 1 and Part 2 to this series.

            Now, we must leave the second scene, and move towards the third and last scene in the painting, and look at those persons who are found in the right foreground, at the bottom of the stairs.

The School of Athens by Raphael (1509 – 1511)

But, we had seen, in part two, that ‘if we look to the right of Diogenes the Cynic and Zeno the Stoic, we see a person, at the very edge of the scene, at the top of the stairs, who is looking over the shoulder of Zeno, at the group of people below and in front of him, at the bottom of the stairs.  But he can’t walk down the stairs to them, because he’s being blocked by Zeno.’  And so, we must search for another way in which to move from this second scene at the top of the stairs, to the third scene at the bottom of the stairs.

            Then, if Antisthenes, Diogenes and Zeno lead to a dead end, and can’t lead us down the stairs, let us retrace our steps back to Plato’s students and his successors – back to Speusippus and Xenocrates.

Xenocrates (left) and Speusippus (right)

            And now, if we look in front of Speusippus, we will see someone who is beginning to go down the stairs – with one foot on the top of the stairs and the other foot on the first step, and who is also looking back, and with his right hand is pointing back, towards Plato.  This should be Theaetetus.

Theatetus (right) and Eudoxus (left)

And, if we look in front of Theaetetus, we see someone who also appears to be walking down the stairs – with one foot on the first step and the other foot on the second step, and is  talking to Theaetetus, and is pointing with both his hands at the person who is sitting down on the steps.  This should be Eudoxus.

And, in front of Theaetetus and Eudoxus, is someone who is sitting down and is spread out over the three steps, with one foot on the second step and the other foot on the third step, and who is reading from a page or a book.  This should be Euclid.

Euclid

The following is from Proclus:

‘Not much younger than these [pupils of Plato] is Euclid, who put together the ‘Elements’, arranging in order many of Eudoxus’ theorems, perfecting many of Theaetetus’, and also bringing to irrefutable demonstration the things which had been only loosely proved by his predecessors.’

The following is from Plato’s ‘Theaetetus’ dialogue:

THEODORUS: Yes, Socrates, I have met with a youth of this city who certainly deserves mention, and you will find it worthwhile to hear me describe him.  If he were handsome, I should be afraid to use strong terms, lest I should be suspected of being in love with him.  However, he is not handsome, but – forgive my saying so – he resembles you in being snub-nosed and having prominent eyes, though these features are less marked in him.  So I can speak without fear.  I assure you that, among all the young men I have met with – and I have had to do with a good many – I have never found such admirable gifts.  The combination of a rare quickness of intelligence with exceptional gentleness and of an incomparably virile spirit with both, is a thing that I should hardly have believed could exist, and I have never seen it before.  In general, people who have such keen and ready wits and such good memories as he are also quick-tempered and passionate; they dart about like ships without ballast, and their temperament is rather enthusiastic than strong, whereas the steadier sort are somewhat dull when they come to face study, and they forget everything.  But his approach to learning and inquiry, with the perfect quietness of its smooth and sure progress, is like the noiseless flow of a stream of oil.  It is wonderful how he achieves all this at his age.

SOCRATES: That is good news.  Who is his father?

THEODORUS: I have heard the name, but I do not remember it.  However, there he is, the middle one of those three who are coming towards us.  He and these friends of his have been rubbing themselves with oil in the portico outside, and, now they have finished, they seem to be coming this way.  See if you recognize him.

SOCRATES: Yes, I do; his father is Euphronius of Sunium, just such another as his son is by your account.  He was a man of good standing, and I believe he left a considerable fortune.  But I don’t know the lad’s name.

THEODORUS: His name is Theaetetus, Socrates, but I fancy the property has been squandered by trustees. Nonetheless, liberality with his money is another of his admirable traits.

SOCRATES: You give him a noble character.  Please ask him to come and sit down with us.

The following is from Pappus:

‘The aim of Book X of Euclid’s treatise on the ‘Elements’ is to investigate the commensurable and the incommensurable, the rational and irrational continuous quantities.  This science has its origin in the school of Pythagoras, but underwent an important development in the hands of the Athenian, Theaetetus, who is justly admired for his natural aptitude in this as in other branches of mathematics.  One of the most gifted of men, he patiently pursued the investigation of truth contained in these branches of science and was in my opinion the chief means of establishing distinctions and irrefutable proofs with respect to the above mentioned quantities.  For Theatetus had distinguished square roots commensurable in length from those which are incommensurable, and who divided the more generally known irrational lines according to the different means, assigning the median line to geometry, the binomial to arithmetic and the apotome to harmony, as stated by Eudemus.’

The following is from Diogenes Laertius:

I. Eudoxus was the son of Aeschines, and a native of Cnidos.  He was an astronomer, a geometrician, a physician, and a lawgiver.  In geometry he was a pupil of Archytas, and in medicine of Philistion, the Sicilian, as Callimachus relates in his Tablets; and Sotion, in his Successions, asserts that he was likewise a pupil of Plato; for that, when he was twenty-three years of age, and in very narrow circumstances, he came to Athens with Theomedon the physician, by whom he was chiefly supported, being attracted by the reputation of the Socratic school.  And when he had arrived at Piraeus, he went up to the city every day, and when he had heard the Sophists lecture he returned.  And having spent two months there, he returned home again; and being aided by the contributions of his friends, he sent sail for Egypt, with Chrysippus the physician, bearing letters of introduction from Agesilaus to Nectanabis, and that he recommended him to the priests.

II. And having remained there a year and four months, he shaved his eyebrows after the manner of the Egyptian priests, and composed, as it is said, the treatise called the Octacteris.  From thence he went to Cyzicus, and to the Propontis, in both of which places he lived as a Sophist; he also went to the court of Mansolus.  And then, in this manner, he returned again to Athens, having a great many disciples with him …

III. He was received in his own country with great honours, as the decree that was passed respecting him shows.  He was also accounted very illustrious among the Greeks, having given laws to his fellow citizens, as Hermippus tells us in the fourth book of his account of the Seven Wise Men; and having also written treatises on Astronomy and Geometry, and several other considerable works.

The following is from Lyndon LaRouche – ‘The Truth About Temporal Eternity’, March 1994:

‘One drives a logical construction to beyond its limit, in the most rigorous way possible, searching for a devastating, axiomatic quality of ontological paradox in those extremes of vastness or smallness.  Once such a paradox is provoked into appearing, the Eudoxian “method of exhaustion” by means of which the paradox is evoked, is examined from the standpoint of the solution prin­ciple of Plato’s Parmenides.’

The following is from Lyndon LaRouche – ‘How Bertrand Russell Became an Evil Man’, July 1994:

‘Insofar as records exist, the more rigorous proof of existence of a class of magnitudes not congruent with rational numbers was developed by Plato’s Academy, following the lines of prior work by Pythagoras et al.  As the geometric proof of the famous Pythagorean theorem is exemplary of this conception, there exists a class of magnitudes in geometry which cannot be rendered congruent with rational numbers: incommensurable magnitudes, such as the hypotenuse of a right triangle.  However, by use of the principle of geometric proportions, we can place these incommensurables between two magnitudes which are congruent with rational number orderings, showing that the incommensurable is less than the one of this pair, but greater than the other.  This concept was embedded in a tactic employed by Plato’s student and collaborator Eudoxus, the “Eudoxian method of exhaustion” which was used by him, and other Greeks to perform an early form of integration, treating the incommensurable volume of a pyramid or cone, for example, as a subject.  Archimedes used this Classical Greek method of Plato’s Academy for his theorems on quadrature.’

Euclid wrote the following books which have survived: Elements, Data, On Divisions, Optics, and Phaenomena, and those books that have been lost: Surface Loci, Prisms, Conics, Book of Fallacies, and Elements of Music.

In the 3rd century AD, Theon of Alexandria edited ‘Euclid’s Elements’ and is the version that was then studied.

The following is from Proclus:

‘This man lived in the time of the first Ptolemy; for Archimedes, who followed closely upon the first Ptolemy makes mention of Euclid, and further says that Ptolemy once asked if there were a shorter way to study geometry than the Elements, to which he replied that there was no royal road to geometry.  He is therefore younger than Plato’s circle, but older than Eratosthenes and Archimedes; for these were contemporaries, as Eratosthenes somewhere says.  In his aim he was a Platonist, being in sympathy with this philosophy, whence he made the end of the whole ‘Elements’ the construction of the so-called Platonic figures.’

The following is from Lyndon LaRouche – ‘How Hobbes’ Mathematics Misshaped Modern History’, Jan. 1996:

‘Until Bernhard Riemann’s 1854 habilitation dissertation, all those formalities of the classroom mathematics which are generally taught still today, were derived from a model of geometry adopted from Euclid’s Elements.  The materialist and empiricist view of that geometry, was based upon the presumption that the four dimensions of Euclidian-Cartesian space-time, were each and all extended into ‘bad infinity’ without limit, and were extended everywhere, always with perfect continuity.  The materialist version of this, assumed that those four dimensions were supplied to an Aristotelian tabula rasa, the newborn human mind, by the human senses, whose sense-impressions were presumed to be a reflection of the composition of the material universe outside the human mind itself.  The empiricists made more limited claims respecting the alleged reality of sense-perceptions, but shared with the materialists the presumption that all knowledge was limited to those ‘facts’ attributed to the self-evident authority of isolable sense-impressions. 

In the real world, which exists only outside such presumptions of Aristotelean virtual reality, the increase of the potential relative population-density of the human species, from the level of a putative man-ape several millions living individuals at most, to the vastly higher population-levels and life-expectancies of civilized existence, is the result of categories of ideas which violate the empiricist’s and materialist’s presumptions respecting sense-perceptions, and respecting ideas as defined by Plato.’ 

The following is from Lyndon LaRouche – ‘Behind the Notes’, March 1997:

What Art Must Learn From Euclid

The possibility of modern science depends upon, the relatively perfected form of that Classical Greek notion of ideas, as that notion is defined by Plato.  This is exemplified by Plato’s Socratic method of hypothesis, upon which the possibility of Europe’s development depended absolutely.  What is passed down to modern times as Euclid’s geometry, embodies a crucial kind of demonstration of that principle; Riemann’s accomplishment was, thus, to have corrected the errors of Euclid, by the same Socratic method employed to produce a geometry which had been, up to Riemann’s time, one of the great works of antiquity.

Look, then, at geometry, again.  Look at Euclid’s geometry of space-time, as the necessary predecessor of its supersessor, the latter the Riemannian geometry of physical space-time.  Focus upon the role of metaphor, as we have considered this, above, in three structured contexts: from the standpoint of a Socratic view of the origins and development of a Euclidian geometry; from the standpoint of a Riemannian geometry; and, from the standpoint of what we have indentified, summarily, above, as the transition from the Euclidean to the Riemannian world-outlook.’

            Having now succeeded in going down the stairs, by following Theaetetus, Eudoxus and Euclid, we have arrived at the scene in the right foreground. 

Now, while looking in the direction that Euclid is facing, we see four young people standing or kneeling around an older person who is bent over and holding a compass up to a diagram on a slate board, as if he is demonstrating a geometrical problem to these four young people.  This should be Archimedes.

Archimedes drawing with a compass

And, if we look behind and to the right of Archimedes, we find someone with a beard, who is wearing a hat, and who is standing and holding in his hand a sphere with the stars marked on it, as if he is demonstrating an astronomical problem.  This should be Aristarchus.

            And, if we look in front and to the right of Archimedes, we find someone standing, facing Aristarchus, with his back to us, and who, in his left hand, is holding a globe of the earth.  This should be Eratosthenes.

Aristarchus holding the globe of stars and Eratosthenes holding the globe of Earth

The works of Archimedes which have survived are: On the sphere and cylinder, Measurement of a circle, On conoids and spheroids, On spirals, On the equilibrium of planes, The sand-reckoner, Quadrature of the parabola, On floating Bodies, Book of lemmas, The cattle-problem, and The method of Archimedes.

The following is from Plutarch – The Life of Marcellus:

‘Yet Archimedes possessed so high a spirit, so profound a soul, and such treasures of scientific knowledge, that though these inventions had now obtained him the renown of more than human sagacity, he yet would not deign to leave behind him any commentary or writing on such subjects; but, repudiating as sordid and ignoble the whole trade of engineering, and every sort of art that lends itself to mere use and profit, he placed his whole affection and ambition in those purer speculations where there can be no reference to the vulgar needs of life; studies, the superiority of which to all others is unquestioned, and in which the only doubt can be whether the beauty and grandeur of the subjects examined, of the precision and cogency of the methods and means of proof, most deserve our attention.  It is not possible to find in all geometry more difficult and intri­cate questions, or more simple and lucid explanations.  Some ascribe this to his natural genius; while others think that incredible effort and toil produced these, to all appearances, easy and unla­boured results.  No amount of investigation of yours will succeed in attaining the proof, and yet, once seen, you immediately believe you would have discovered it; by so smooth and so rapid a path he leads you to the conclusion required.  And thus it ceases to be incredible that (as is com­monly told of him) the charm of his familiar and domestic Siren made him forget his food and neglect his person, to that degree that when he was occasionally carried by absolute violence to bathe or have his body anointed, he used to trace geometrical figures in the ashes of the fire, and diagrams in the oil on his body, being in a state of entire preoccupation, and, in the truest sense, divine possession with his love and delight in science.  His discoveries were numerous and admi­rable; but he is said to have requested his friends and relations that, when he was dead, they would place over his tomb a sphere containing a cylinder, inscribing it with the ratio which the containing solid bears to the contained.

Archimedes, however, in writing to King Hiero, whose friend and near relation he was, had stated that given the force, any given weight might be moved, and even boasted, we are told, relying on the strength of demonstration, that if there were another earth, by going into it he could remove this.’

The following is from Pappus:

‘Although many solid figures having all kinds of surfaces can be conceived, those which appear to be regularly formed are most deserving of attention.  Those include not only the five figures found in the godlike Plato, but also the solids, thirteen in number, which were discovered by Archimedes and are contained by equilateral and equiangular, but not similar, polygons.’

The following is from Lyndon LaRouche – ‘How Bertrand Russell Became an Evil Man’, July 1994:

‘Although the roots of modern science are found in Plato’s Academy of Athens, modern science as such began with Nicolaus of Cusa’s ‘De Docta Ignorantia’, published in the setting of the 1439 – 1440 Council of Florence.  It was Plato’s Academy which first supplied a rigorous treatment of the problem of the ‘immeasurably small’.  The central formal feature of Cusa’s breakthrough in mathematical science was his application of the solution-principle of Plato’s Parmenides to effect a correction in Archimedes’ constructive efforts at quadrature of the circle.  Cusa’s work bears directly on the issue of the same ‘immeasurably small’.  This case bears directly upon the central fraud of Russell’s work in mathematics, a fraud which is also central to all radical empiricism and its positivist derivatives.’

The surviving work of Aristarchus is On the Sizes and Distances of the Sun and Moon.  The book referred to by Archimedes has been lost.

The following is from Archimedes – ‘The Sand-Reckoner’:

‘Now you (King Gelon) are aware that ‘universe’ is the name given by most astronomers to the sphere whose centre is the centre of the earth and whose radius is equal to the straight line between the centre of the sun and the centre of the earth.  This is the common account as you have heard from astronomers.  But Aristarchus of Samos brought out a book consisting of some hypotheses, in which the premises lead to the result that the universe is many times greater than that now so called.  His hypotheses are that the fixed stars and the sun remain unmoved, that the earth revolves about the sun in the circumference of a circle, the sun lying in the middle of the orbit, and that the sphere of the fixed stars, situated about the same centre as the sun, is so great that the circle in which he supposes the earth to revolve bears such a proportion to the distance of the fixed stars as the centre of the sphere bears to its surface.’

The following is from Plutarch – ‘On the Face of the Disc of the Moon’:

‘Only do not, my good fellow, enter an action against me for impiety in the style of Cleanthes, who thought it was the duty of the Greeks to indict Aristarchus of Samos on the charge of impiety for putting in motion the Hearth of the Universe, this being the effect of his attempts to save the phe­nomena by supposing the heaven to remain at rest and the earth to revolve in an oblique circle, while it rotates, at the same time, about its own axis.’

The following is from Cicero – ‘The Republic’, Book 1:

‘He (Gallus) ordered the celestial globe to be brought out which the grandfather of Marcellus had carried off from Syracuse, when that very rich and beautiful city was taken, though he took home with him nothing else of the great store of booty captured.  Though I had heard this globe mentioned quite frequently on account of the fame of Archimedes, when I actually saw it I did not particularly admire it; for that other celestial globe, also constructed by Archimedes, which the same Marcellus placed in the temple of Virtue, is more beautiful as well as more widely known among the people.  But when Gallus began to give a very learned explanation of the device, I con­cluded that the famous Sicilian had been endowed with greater genius that one would imagine it possible for a human being to possess.  For Gallus told us that the other kind of celestial globe, which was solid and contained no hollow space, was a very early invention, the first one of that kind having been constructed by Thales of Miletus, and later marked by Eudoxus of Cnidus (a disciple of Plato, it was claimed) with the constellations and stars which are fixed in the sky.  He also said that many years later Aratus, borrowing this whole arrangement and plan from Eudoxus, had described it in verse, without any knowledge of astronomy, but with considerable poetic talent.  But this newer kind of globe, he said, on which were delineated the motions of the sun and moon and of those five stars which are called wanderers, or, as we say, rovers, contained more than could be shown on the solid globe, and the invention of Archimedes deserved special admiration because he had thought out a way to represent accurately by a single device for turning the globe, those various and divergent movements with their different rates of speed.’

None of the works of Eratosthenes have survived.  ‘Platonicus’ and the history of the problem of duplicating the cube, can be found in ‘Exposito rerum Mathematicarum’ by Theon of Smyrna.  The ‘Seive of Eratosthenes’ can be found in ‘Introduction to Arithmetic’ by Nicomedes.  ‘On the measurement of the Earth’ can be found in ‘De Contemplatione Orbium Ceolestium’ by Cleomedes.  ‘On Means’ is mentioned by Pappus as one of the great books of geometry.

The following excerpt is from ‘Who Was Erathostenes’ by Bruce Director, a subsection to ‘On Eratosthenes, Maui’s Voyage of Discovery, and Reviving the Principle of Discovery Today’ by Lyndon LaRouche:

‘Eratosthenes, perhaps the greatest scientist of the Hellenistic world, was also one of its most prolific and versatile: His work included investigations in astronomy, geography, geodesy, poetry, music, drama, and philosophy.  Born in Cyrene, he was educated in Alexandria, Egypt, and Athens by followers of Plato.  At the age of 40, he became the head of the famous library at Alexandria, where he remained until his death.  In addition to his measurement of the Earth’s circumference, Eratosthenes was the first to measure the angle of the Earth’s tilt on its axis (the plane of the eclip­tic).  He also wrote ‘The Duplication of the Cube’, and ‘On Means’, which were treatises investigating the crucial mathematical paradoxes arising from the investigation of dimensionality.  His work ‘Platonicus’ deals with the mathematical and musical principles of Plato’s philosophy.  He published maps and works on geography and chronology.  Eratosthenes was also a poet, dra­matist, and philologist, writing several poems and plays, only fragments of which survive, and a book on comedy.  Other ancient writers attribute to Eratosthenes books on philosophy and history.’

The following is from Lyndon LaRouche – ‘The Essential Role of Time Reversal in Mathematical Economics’, October 1996:

‘We begin at a point which leads most directly to the fundamental discovery of principle set forth in Riemann’s 1854 habilitation dissertation: the celebrated measurement of the curvature of our planet by Eratosthenes.  In recent time, I have often employed this discovery by Eratosthenes.  That choice reflects the fact that this discovery provides the simplest, cleanest example of the way in which Platonic ideas arise in every fundamental, experimental discovery of physical principle.  By comparing the angles cast by the noonday shadow upon the interior of hemispherical sundials, along the meridian linking Syene (Aswan) to Alexandria, in Egypt, Eratosthenes demonstrated, geodetically, that the Earth was a spheroid, estimating the Earth’s polar diameter with a margin of error of approximately fifty miles.  The relevant paradox is, that Eratosthenes measured the curva­ture of the Earth’s meridian more than two thousand years before any person was to have seen our planet’s curvature.  The principle of the Earth’s curvature, as adduced thus, represents a Platonic idea: a conception of measurable relationship, a relationship which is not directly perceived as a sense perception, nor as a new theorem of an existing deductive form of theorem-lattice. All such notions of measurable relationship which underlie the principles of astrophysics, are obtained only as ‘Platonic ideas’.

The Geometric Proof

Let us look again, at how we moved from the second scene to this third scene: from the standing Theaetetus and Eudoxus, to the sitting (almost prostrate) Euclid.  At the bottom of the stairs, however, we have moved from a half-kneeling student, through the bent-over Archimedes, to the standing Aristarchus and Eratosthenes.

We are reminded of the fate of Athens: as the New Academy (265 B.C.) is falling into skepticism, some Greeks leave Athens and move to Alexandria, Egypt to revive the real Plato (such as Eratosthenes who became head of the library there).

In looking at this scene and comparing it with the previous two scenes (the first – the Pre-Socratic ‘Paradox’ and the second – Plato’s ‘Solution-Principle’), it appears that each of the three persons here at the bottom of the stairs – Archimedes, Aristarchus and Eratosthenes – lived after the time of Plato, and are seen being involved in showing or demonstrating something, as if trying to ‘prove’ something – to ‘prove’ a ‘Platonic idea’.

The Pre-Socratic ‘Paradox’. Refer to Part 1 of this series.
Plato’s ‘Solution-Principle’. Refer to Part 2 of this series.

And, we have arrived at the final stop of our ‘three-step method of sharing such an experience – paradox, hypothesis and validation’!

The Geometric Proof

The following is from Lyndon LaRouche – ‘That Which Underlies Motivic Thorough-Composition’, Aug. 1995:

‘Today’s generally accepted university-classroom mathematics, finds its origins in a creation of the naive imagination, in an image of space-time like that offered by a traditional classroom reading of Greek geometry.  In that naive fantasy, space is defined axiomatically in terms of three primary senses of direction, which are assumed to be extensible, both without limit, and with perfect continuity: backward-forward, up-down, and side-to-side.  To time is attributed a single sense of direction: backward-forward. 

The principal postulates of that notion of quadruply-extended space-time, whose magnitude is absolute zero, and that a ‘straight line’ is the shortest distance between two points in space.  These postulates are required by the axioms of the trebly-extended space manifold.  Neither sense-certainty, nor such a mathematics makes any provision for the existence of cause within our universe. 

The attempt to develop a mathematical physics consistent with that naive sort of quadruply-extended space-time manifold, consists of mapping the location of the points within an object such that those correspond to points in naively defined space.  Change of that mapping, with respect to time, is assumed to represent a linear form of motion.  Forms of change other than simple displacement in space-time, are defined naively in terms of the simple idea of motion.  No provision for cause is supplied. 

That species of naive mathematical physics comes into crisis when experimental evidence presents forms of motion, and related change, which cannot be accounted for in terms of the axiomatic features of naive space-time.

This was already noted by leading fig­ures of Plato’s Academy of Athens, and their followers, such as Aristarchus, Archimedes, and Eratosthenes.  For example, simple astronomy showed that measurements on the surface of the earth required a spherical geometry, rather than a plane geometry.’

The following is from Lyndon LaRouche – ‘The Coming Death of Systems Analysis’, March 2000:

‘The discoveries of what are later experimentally validated as universal physical principles, are prompted by the demonstration of those qualities of paradoxes, the which are not susceptible of formal solution by means of the deductive and other methods of the philosophical reductionists.  Such paradoxes are typified by the ontological paradox of Plato’s Parmenides dialogue; the impossibility of solving such by deductive methods, is typified by the case of that historical Parmenides, whose method Plato referenced in that dialogue. 

A successful solution is generated when something occurs, the which is sometimes described as an ignited flash of insight, to produce a validatable hypothesis in that person’s mind.  The acceptance of that hypothesis by other persons within society, requires that two special conditions be satisfied.  First, the same experience of insight must be replicated, independently, within the sovereign cognitive precincts of at least one other individual’s mind.  Second, that hypothesis, so generated, must be shown to be an existent, efficient principle, by means of experimental demonstration of the efficiency of its willful application to the physical domain as a whole.  The latter such experiments belong to the class which Riemann defined as unique: it is not sufficient to show experimentally that the prescribed effect might be produced; it must also be demonstrated that that hypothetical universal principle coheres, in a multiply connected way, with all validated other universal physical principles.

The crucial point is, that the only way in which we can generate a functionally efficient notion of such a cognitive idea existing in another mind, is the three-step method of sharing such an experience (paradox, hypothesis, validation), as I have just identified this summarily.  In such a case, we know three essential things.  First, we know, independently of our cognitive processes, the paradox which prompted the generation of a discovery of principle, as the only feasible solution to that paradox.  Thirdly, we know the manifest experimental proof of the proposed solution.  Thus, by sharing the first and third of those steps, we are able to correlate the specific act of cognition, the second step, in the other mind, with that recallable experience of cognition we experience in our own.

Conclusion

Now, let us look back over the path we have taken in traveling through the symmetry of the whole painting.  At first, we saw the two statues, of Hephaestus and Athena, the founders of the city of Athens.  And then, we saw our opening bookend – our announcer of the play, Diogenes Laertius, and three persons around him.

In walking through the three scenes, of those who lived before Plato, of those who lived at the time of Plato, and of those who lived after Plato, we saw and named twenty-six Greek philosophers, and their relation to Plato, the founder of the Academy – the School of Athens, and the author of the Timaeus (the book at the center of the painting – refer to Part 2 of this series). 

In walking through those three scenes again, we saw but did not identify another twenty-six people:

– the four persons in the first scene (1 person holding the slate for Pythagoras and looking at a 2nd person, and two persons, in white, who are passing through the scene and also are looking out at us, the viewer);

–  the eighteen persons in the second scene (2 persons at each end of the scene, and the sixteen students of Plato);

– and the four students of Archimedes in the third scene.

We now come to the last two figures in the painting – our closing bookend. 

If we look to the right of Aristarchus and Eratosthenes, we see two other people who are also standing there. The first figure we know to be Raphael, who painted himself into the picture, and thus, draws our attention to this final part of this scene.  Now, to the right of Raphael, is someone standing, wearing a white hat and a white cloak.  Both Eratosthenes, with his globe, and Aristarchus, with his celestial globe, are both looking at ‘him’, while he is looking back at them, as if he was in their time and place, and discussing with them the estimate of the circumference of the earth and how the earth moves about the sun.  Yet, he’s dressed and wearing a hat similar to Raphael’s, as if they were contemporaries !?! 

Who could be living in two different time periods?  Who could be living at two different places?  Does the simul­taneity of eternity really exist?  Who could this person be?  Someone, recently deceased, who should be remembered, perhaps?  The only person that I could adduce, or guess at, would be Christopher Columbus.

Raphael (inner right) looking out at us the audience and Christopher Columbus ( far right).

The following is from Lyndon LaRouche – ‘Columbus’s Discovery of America and the Strategic Crisis Today’, June 1992:

‘What was Columbus?  Well, he was a man, of course.  You have his history from other sources at this meeting.  But what was he essentially, in terms of his role of discovery in the Americas?  He represented an institutional force with two aspects, which centered around the work of the Golden Renaissance from the third and fourth decades of the fifteenth century.  He represented those around Cusa, including Paolo dal Pozzo Toscanelli, and others, who had a new conception of society reflected in part by Cusa’s earlier work.  Cusa’s key work is ‘Concordantia Catholica’: a new conception of society and a new political conception of man in terms of society.  Oh, it was in the Christian tradition and the Apostolic tradition, and it was in the tradition of Augustine and therefore also of Aquinas; it was a Platonic, anti-Aristotelian conception of man.  And Aquinas was really anti-Aristotelian, though many will dispute that for various reasons. 

So this group of people did something which had two facets.  One, was to develop the science- and I understand you have a report on some of the science behind Columbus’s discoveries.  These people determined approximately how large the world was, by simple but obvious methods.  We may say after looking at them, that they seem to be obvious today.  They’re obvious kinds of astronomical calculations, which told them how large the planet is, based on the fact that the planet rotates over approximately twenty-four hours; and then use that kind of information, and the changes in the declination of the Sun and this sort of thing, and the ocean and some of the planets, to estimate how large the world was.  A very good job, as a matter of fact.  They constructed estimated maps of the world.  They built a science which enabled the navigation to occur, which discovered the Americas.’

The following is from Lyndon LaRouche – ‘Why We Must Colonize Mars’, November 1996:

‘Columbus’s discovery of the Americas began toward the close of the third century BC, with the estimate of the Earth’s curvature by the celebrated member of the Platonic Academy at Athens, Eratosthenes.  Employing Eratosthenes’ and other ancient experiments as his guide, Paolo Toscanelli (1397-1482), the leading astronomer of the fifteenth century, created the maps of the world which guided Columbus to his successful voyage.  Toscanelli’s map had but one notable flaw; it was based upon a nearly accurate size of the Earth, as determined by astronomical observations of the Earth’s curvature, but, it relied upon the highly exaggerated reports supplied by Venice, on the distances from Venice to China and Japan, placing Japan and the islands of the Indies in the middle of today’s United States! 

Columbus learned of Toscanelli’s maps nearly two decades before his famous voyages of discovery.  This included Columbus’s access to the correspondence between Toscanelli and Lisbon’s Fernao Martins, on the subject of exploration westward across the Atlantic Ocean to the Indies.  Columbus wrote to Toscanelli and became fully informed, in the last years of Toscanelli’s life, of the collaboration which had been ongoing for decades before, and which had begun with the immediate Florentine circle of Nicholas of Cusa during the years before the Council of Florence of 1439.  Columbus added to this scientific knowledge, his experience and knowledge as a navigator for the Portuguese, knowledge of ocean currents and prevailing winds, which clearly implied the probable location of, and route toward land on the other side of the Atlantic.  His use of Toscanelli’s map, indicates that his original goal were the islands of the Pacific far to the south of Japan.  Columbus’s discovery of the Americas was, thus, a ‘scientific discovery’, in the strictest meaning of experimental physics.’

But now, looking back again, starting with Thales’s left foot on a block, and then Pythagoras’s left foot on a block, that are seen as the stepping stones to begin our journey, we arrive at Columbus.  And there to the left of Columbus, we see one other block – but no one has their foot upon it.  Not yet.

If we look back once more, at Raphael, we notice that he is looking out of the painting – at us.  And, if we walk through the painting once again, wherever we are, Raphael is still looking at us.  So that now, if you look at all the people in the painting, living at different times, in different scenes, and all at once, and with Raphael still looking at you, you then realize that there is one more person in this painting – you, the viewer. 

The question now posed is, ‘Who will come forward to place their foot on this block – this stepping stone on our new future journey?’

The following is from Lyndon LaRouche – ‘The Truth About Temporal Eternity’, March 1994:

“Stand in the old papal apartments, now part of the Vatican museum.  Stand facing the famous ‘School of Athens’, a subject on which a bit has been said here already.  The reasons you must be there in Rome to receive in full the message being sent personally to you across nearly five hundred intervening years, should be obvious to anyone who sees it there.  In the meantime, as very few of you are presently visiting that Museum, concentrate upon any of the better reproductions of this mural; the less the reproduction in scale, relative to the original, the better for our purposes here.  It will help you to situate yourself mentally, as if you were actually standing in that great hall depicted there.  As you stand there, call that mural to life.  Look around inside that mural; which of these are old friends of yours?  You never met any of them face to face, but most of those in the hall never met one another in the flesh, either.  Yet, you have relived a most intimate moment of the mind of each of some of them, reliving one or more of their creative moments of discovery. 

First, pick those whom you know in that way.  You know Plato, and are acquainted with Aristotle.  Are there not two or three in the foreground?  As you focus upon the ideas, especially those ideas which represent original axiomatic-revolutionary discoveries, or something proximate to that, one figure after another within this busy hall comes alive for you.  As for the others, I believe you know most of them by reputation.  Think of the number of generations of history spanned by the personalities gathered here within this hall!  Radiating from that hall, there is a sense of being embraced, where you stand, by some living intelligence proximate to Temporal Eternity.  That radiance fills the small room in the old papal apartments.  Raphael understood the point well enough to design and transmit a message, this mural, which would reach both of us, nearly five centuries later, standing with our minds within that mural’s assembly within the great hall.  It is no fantasy; it is a painting of a scene the like of which this writer has seen within his own mind, many times.  It is a scene which Raphael painted from life, with the gathering of the inhabitants of his mind as living models.  It draws from life those relationships within Temporal Eternity which are higher, and more efficient than any drawn in ordinary space or ordinary time. 

Those are the direct relationships of creative minds’ ideas, which dissolve centuries into the span of a pleasant day’s assembly, and bring vast spaces comfortably into a room no larger than that which contains this mural.  This mural is no mere symbolism, nor an imagined room in Paradise.  It is a moment of deja vu!  It is a portrait of Raphael’s relations to the most intimate acquaintances of his daily mental life, all captured so to share the companionship of a moment in Temporal Eternity.  That mural is also a religious experience.  When the social reality of Temporal Eternity compacts centuries into a morning’s gathering in such a fashion, the universe of time and space is shrunken to such a smallness that we seem almost to wrap it all within our mind.  In such a circumstance, we are compelled to hypothesize higher hypothesizing in such a way, that an eerie sense of a timeless Absolute Intelligence’s efficiency is aroused within us. When the relationship of the individual person to mankind in general, and other persons in particular, is measured in the space and time of the generation and transmission of those qualities of ideas associated with valid axiomatic-revolutionary discoveries, what a short distance a mere few centuries becomes!  The order of necessary predecessor and necessary successor is preserved: The intelligence of the timeless Absolute is not zero-motion; the lack of spatial division is the consequence of being simultaneously everywhere, such that there is nothing in between any two experiences which would require us to experience time, except as, for us the onlookers, a sense of timeless ordering of development. 

For us, the onlookers, just so, the duration of space and extent of time shrink almost to the vanishing-point.  So, if the mind of any among us is sufficiently developed to grasp the transmission of a valid axiomatic-revolutionary discovery, effected by one person, to cause the reliving of that act of discovery of that conception in the mind of a single person hundreds of years, or even millennia later, whoever has gained those qualifications is able to see the world as that mural portrays its more essential features.  Once that step is made, he or she is able to see the essential relations of humanity as Raphael portrays that viewer’s relationship to his ‘School of Athens’.”

And, so let us end our journey through the discovery of the ‘School of Athens’, with a short except of an address by the ‘poet-president’ Abraham Lincoln:   

       ‘… It is for us the living, rather, to be dedicated here

                                    to the unfinished work which they who fought here

                           have thus far so nobly advanced.

                                    It is rather for us to be here dedicated

                           to the great task remaining before us –

                                    that from these honored dead

                           we take increased devotion to that cause

                                    for which they gave the last full measure of devotion –

                           that we here highly resolve

                                    that these dead shall not have died in vain –

                           that this nation, under God,

                                    shall have a new birth of freedom –

                           and that government of the PEOPLE,

                                    by the PEOPLE,

                           for the PEOPLE,

                                    shall not perish from the earth.’

For more information on the poetic principle, read Why the Poetic Principle is Imperative for Statecraft, and watch the Rising Tide Lecture Series ‘Towards an Age of Creative Reason

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