Compiling information

Simplistic tales are often those hardest to refute, not because they are true, but because they form a clear and understandable picture of the messiness of the real world. Take the neat story of how the ancient world turned to Middle Ages: Christians took over the kingdom of Rome, persecuted all the pagans doing real science, burned their writings and killed them, making Europe into a cultural wasteland, soon made into a real wasteland by the wandering barbarian tribes.

This particular story makes an essential mistake in supposing that Christians would have had any unified agenda. Instead, we have on the level of church governance, many sects competing on what the official dogma should be, and even within the sect that finally became the Catholic church, there were many actors with diverse agendas. Individual outbreaks of violence against polytheists by Christians did occur - most famous being the murder of the mathematician and philosopher Hypatia - but these were not officially sanctioned and often these were even condoned as not in the spirit of Christianity.

The church wasn’t completely neutral toward polytheists: bishops were happy to hinder all public display of traditional polytheistic rites by removing altars for the old gods, and some books on magical rituals were ordered to be destroyed. Yet, the prevalent tendency toward classical culture, literature and learning was one of appreciation.

The main reason for the destruction of many of the ancient pieces of writing is simply that physical copies of them were not eternal - the material that they were written on might have deteriorated or it might have been cleaned of text for further use - and there was no long lasting attempt to preserve them all. If there weren’t a large number of copies of some work in circulation, it was highly probable that it wouldn’t survive into modern days.

Lot of classical works did survive in the eastern part of the Roman empire, but general histories of European thought often tend to forget this, since they are more interested in the developments in the western side of the empire. The two parts were, of course, differentiated by language, Greek being the major language in the East, Latin in the West. As we have noticed many times earlier, Latin part of the Roman empire was almost like an intellectual backwater, where ideas discussed in the Greek part flowed much later and in not so great abundance. When the two parts drifted finally apart and the communication between them diminished, the Latin-speaking part had on many fields of learning only few summarised compilations to lean on to.

Compilers like Martianus Capella were thus important for the development of western medieval thought, even if the content of their works was far from groundbreaking. Martianus’ work was also a literary example for many medieval allegories. His compilation is framed by a story of god Mercury, patron of rhetoric, marrying a mortal woman, Philology, representing scholarly expertise. The setting of the story combines traditional polytheism with Neoplatonic influences - mythological stories of gods are joined with a more philosophical account of divinely animated planets, and Jupiter especially plays also a role of the Neoplatonic Intellect or Nus. Particularly noteworthy is the idea that even humans can rise to the celestial rank of gods, if they just educate themselves enough.

The story of the marriage leads to the handmaidens of Philology, each named after some field of learning, giving a speech about her field. Somewhat surprisingly, given that their audience is supposed to be full of learned divinities, the content of these lectures is rather elementary. Thus, the first handmaiden, Grammar, speaks about Latin letters, possible syllables formed of them and different classes of inflections. The second handmaiden, Dialectics, speaks then of the semantic classes of the words, continues by indicating how words are combined into sentences, shows various simple logical relations between sentences and finishes by showing what new sentences can be deduced out of given assumptions. The third handmaiden, Rhetorics, indicates various situations, where sentences should be put together into convincing speeches, delineates the structure of such speeches and rounds off with tricks for making the speech sound eloquent.

Martianus’ account of the more mathematical disciplines is even less impressive. The fifth handmaiden, Geometry, spends most of her time telling about the geography of the known world, on the pretext that geometry is all about measuring land areas. Of the discipline proper, she manages to expound only few elementary definitions of basic shapes. Arithmetic fares a bit better, as she uses only relatively little time with Pythagorean numerology and then proceeds to expound with comprehensive detail some basic properties of numbers and their divisibility. Even so, she does not really prove any arithmetical theorem she presents, but merely exemplifies them with some appropriate numbers.

The final chapters of the book concentrate mostly on some elementary facts of the disciplines in question. The account of Astronomy seems to contain no mention of any astrology, although we cannot be sure, since what has survived from her speech lacks the ending. Harmony does begin her speech by telling how the music guides the heavenly spheres, but then quickly continues with basics about rhythm and melody.

While Martianus’s book is an independent work, specifically dedicated to presenting basic information on a systematically chosen set of fields of learning, achievements of Greek scholars were transmitted to the early western medievals also through commentaries on works of philosophers from previous generations. As the prevailing character of the philosophy of the day was Platonist, so the philosophers chosen were also of Platonist schools of thought. So, while Martianus presented learning with a Platonist covering, these commentaries were completely soaked in Platonist teachings.

We’ve already seen one example of such a commentary, namely Calcidius’ work on Plato’s Timaeus, which contained, for instance, a section dedicated to astronomy. Another work of the same sort was written by Macrobius. He chose to comment on a work of Cicero on good republic. Cicero’s obvious reference point had been Plato’s seminal work on republic, and just like Plato had ended his work with a fable showing what happens to people after death, Cicero had also finished with a section describing how Scipio, a Roman general and statesman, had seen a dream describing what life after death was like.

It was this specific section in Cicero’s work that Macrobius wanted to deal with. Because ironically at least from Plato onward philosophers had been doubtful of the worth of doing philosophy through fables, Macrobius sees it fit to begin his commentary with an apology of philosophical fables. Macrobius notes that the highest levels of the ontological hierarchy cannot be dealt in fables, since these are completely detached from the world of perception and cannot be clothed in images, except through some inappropriate analogies. Then again, they are appropriate, he says, specifically, when dealing with divinities of common polytheism and the world-soul, because they are closely connected with the world of perception, but higher in status than mere earthly things and therefore should not be discussed openly, but with a reverent attitude.

Because Cicero’s work is a fable, and even more, a fable about a prophetic and revelatory dream, Macrobius appears to think that he can look through the external account of the story and find more substantial truths even in its minor details. This makes Macrobius’ commentary appear to be full of digressions not really connected with Cicero’s work. For instance, when Scipio is told by a soul of his dead relative that a certain event will occur to him when numbers determining his life are full, Macrobius takes this as an opportunity to explain Pythagorean numerology.

Although Macrobius’ account of the properties of numbers appears to be rather full of insignificancies, he seems to be particularly interested in the idea that numbers determine the cycles of human life. This does not mean, Macrobius continues, that everything in human life is preordained - some prophecies can be avoided - but certainly some of it is, and everyone will die eventually.

Now, Cicero had noted that what happens to a person after their death is dependent on their way of life, thus, Macrobius says, he had to study ethics. Particularly, Cicero had said that work for one’s commonwealth earned a person a place in heavens. Macrobius is adamant to reconcile Cicero’s statement with a later Neoplatonic idea that political life is not the highest form of life. Macrobius thus sees life dedicated toward the well-being of other people as a stepping stone toward life where one is completely removed from earthly worries.

Indeed, Macrobius is dedicated to the Platonist idea that real death is actually the earthly life where a human soul is imprisoned by the body and its passions and desires, while death as the separation of a soul from its body gives it a chance to return to a disembodied life. This does not mean that a human being should try to actively look for death, Macrobius says, because passion for death would merely lock the soul back to a bodily life.

In addition to ethics, Macrobius says, Cicero studied physics or the structure of the world of perception. Macrobius incorporates the physical world in the common Neoplatonist hierarchy, where the incomprehensible primal source of all being creates an intelligence that both imitates the source by thinking it and in looking away from its source creates the world-soul. The world-soul, again, both imitates the intelligence and merely thinks, but also looks away from it and creates a number of bodies to which it entangles itself, dividing into a number of individual souls.

This is the place where Macrobius introduces his account of astronomy. Lot of the details Macrobius goes through are, of course, same as in Martinus’ account of astronomy, although there are some significant differences (for instance, Martinus upholds a more sophisticated theory where Mercurius and Venus circle around the Sun, not the Earth). Yet, there is an even more important difference in the reason why Macrobius is interested in astronomy. Firstly, it provides a map of the journey the human soul takes when landing from the remotest sphere of the world to Earth, gaining new and more embodied capacities at each planet. Secondly, it shows the greatness of heavens and insignificance of Earth in comparison.

Macrobius continues with an account of the geography of the Earth. His main idea is that the area of lands accessible to Roman civilization is minuscule in proportion to all the inhabited lands on Earth. Connected with the cosmological fact that the heavens circle the Earth eternally and the consequent historical fact that the human civilizations have again and again risen and fallen this idea leads Macrobius to emphasise the insignificance of earthly life.

From physics, Macrobius says, Cicero moved finally to a study of the divine. He especially pointed out the only spark of the divine on Earth, namely, the human souls. The essence of soul, Cicero said, following Plato, lies in self-movement. Macrobius defends this notion against the criticism of Aristotle that mover always differs from moved, that self-movement is therefore impossible and that all movement has begun from an unmoved mover. Macrobius follows earlier Neoplatonists in insisting that self-movement does not mean that the soul would somehow act upon itself, but simply that the soul is by nature active, just like fire does not need something outside itself to become warm. This self-motion of the soul, Macrobius explains, is not literal change of place, but closer to such activities like thinking, emoting and wanting, and through such activities human souls move.their bodies, just like the world soul moves with such activities the stars and planets

What Macrobius finds in Cicero’s work is the whole philosophy. What medievals would find in Macrobius is something else. For them, important things would be the astronomical facts and such interesting details like categories of dreams. The overarching philosophy they would get elsewhere.

Numbering intellect

Like Aristotle before him, Plotinus placed observation above action. Yet, with Plotinus observation was not completely passive, but rather more active than normal action. For instance, Plotinus noted, nature does not act in the sense of mechanically moving things around and does not even consider how to achieve some end, but merely observes its own perfection and at the same time projects energy, which makes physical world move. The movements and activities of the physical world are then less active than this free letting of energy by nature, but it is the best that this world can do.

This nature was in Plotinus' hierarchy something proceeding from the level of the soul, or energy projected by the world soul. On the level of souls, the hierarchy of observation and action was also present. Thus, Plotinus noted, people who are not capable of observing things in their mind and thus engaging in abstract sciences have to perform manual labour. Furthermore, even when people performed things physically and endeavoured after some end, this end was always some type of observation (for instance, we produce something because we want to see it before our eyes). Hence, the priority of observation was evident in Plotinus’ eyes.

While observation was the highest soul could achieve, it was the very essence of intellect in Plotinus’ hierarchy to observe itself and everything within itself. Of course, Plotinus noted that this wasn’t still the highest point in the hierarchy, because the primordial unity wouldn’t even need observation to be perfect - indeed, to be perfect it couldn’t have any multiplicity required even by self-observation and in no way it could contain all that exists, like the self-observation of intellect, although this primordial unity was the final source of everything.

Although the level of self-perceiving intellect and its ideas wasn’t the highest pinnacle in Plotinus’ hierarchy, it was the pinnacle of beauty. Indeed, he noted, an artist does not make beautiful works by choosing the appropriate matter e.g. for sculpture, but by imprinting a beautiful shape, existing already in her mind, to the material of the sculpture. Similarly, Plotinus thought, nature does not make beautiful things by choosing suitable material, but by projecting a beautiful shape on the material - what material does at most is to hinder the projection and lessen the beauty.

The self-thinking intellect then contains all these beautiful prototypes in a single act of self-thinking, thus being the pinnacle of all beauty. This shows us something important about the level of intellect in Plotinus’ hierarchy. The thinking of the intellect is not what we usually call thinking, that is, silent speaking of thoughts in our mind or an argument moving from premises to conclusion, which inevitably takes some time to occur. Instead, it is more like a momentary vision revealing all that is important in a single glance. Hence, intellect does not need to think all the beautiful prototypes one by one, but it at once has them all in front of mind’s eye. Furthermore, Plotinus thought that intellect also viewed these prototypes truly. Indeed, since they were not anything truly separate from the intellect, but merely aspects of the self-observation of intellect, there could be no room for false or incorrect observation of them.

Because of its beauty and truth, Plotinus called self-thinking intellect a god. Indeed, he compared it with Chronus, who held the middle place between his son Zeus (soul) and his father Uranus (primordial unity). According to Plotinus, this middle position made intellect unique and different from both soul and unity in the sense that only intellect found its complete perfection in observation. For “Zeus” or soul mere observation wasn’t enough, but it also regulated the world around it, while “Uranus” or primordial unity did not really need this observation for being perfect.

This primordial unity was something that defied all human characterisations. It had no limits, but it was equally wrong to call it infinitely large. It wasn’t contained in anything, although it in a sense potentially contained everything that ever was actually. One clear characteristic of this unity was its perfection or goodness, which was then something higher than mere beauty. In fact, Plotinus remarked, everything in the world strives for goodness, even when they sleep, while beauty interests only those who can observe things.

Three levels above corporeal world was enough for Plotinus: primordial unity, self-thinking intellect and soul. He was especially against gnostic teaching, where the number of entities was expanded beyond any need. For instance, beyond intellect one did not need any further entity conscious of intellect, because this role in the hierarchy was filled already by intellect itself, which could observe itself.

Gnosticism was to be rejected for other reasons also, although Plotinus noted that many of his friends had been lured by the ideas of gnostics. He was especially against the idea that corporeal world had been created through a mistake and that it was clearly evil in comparison with the spiritual world. Indeed, Plotinus noted that corporeal world was the best possible image of the best possible prototype, necessarily proceeding from the energy of this prototype.

Furthermore, while gnostics regarded the souls of gnostics as the only element of goodness in the corporeal world, Plotinus stated that many things were good and beautiful in it. For instance, stars and planets were in Plotinus’ eyes divinities, which were unencumbered by their bodies and could thus always observe the spiritual prototypes of the world. Even the unconscious nature was full of purpose and goodness and was guided by soul-like unconscious activity of nature.

Plotinus finally noted that gnostic ideas gave no guidelines how a person could improve herself - one was just born as spiritual or as corporeal and there was no escape from one’s fate. Indeed, gnostic could not care less about the fate of the body his soul inhabited, because it was a prison for the soul. Plotinus himself, on the other hand, did not regard embodiment of soul in such a low esteem, but as a necessary step in the development of the soul. Thus, by living with the body soul was meant to learn indifference toward material objects and thus rise to the level of stars.

Gnostics weren’t the only school of thought Plotinus engaged with. He was also quite interested of the Pythagorean question of the status of numbers in the hierarchy of existence. Plotinus noted that one could not simply identify numbers with other things, which we would do, for instance, if we placed existent things in an order and called first thing “one”, second “two” and so on. Such an identification would fail because number like “three” could be used of many things, not just of the third thing in that ontological order. Indeed, numbers had to precede all of these existent things, because one had to have e.g. a prototype of ten that could be applied to all combinations of ten things.

If Plotinus thought numbers were independent of things, he also though they were independent of any acts of calculation. Indeed, this was just a particular application of a general principle that something was in Plotinus’ opinion always prior to thinking of this something - e.g. the existence of movement was prior to thinking about the movement. This was even true of the self-thinking intellect, which contained all ideal prototypes within its act of self-thinking. Indeed, one could distinguish within this act various levels - the self-thinking intellect was e.g. living, but more essentially it was thinking and even more essentially existing. Numbers, then, Plotinus concluded, belonged to the level of existence of intellect. In fact, while primordial unity was for Plotinus something preceding in a sense existence, numbers preceded, according to him, the multiplicity of existents - one could say that while primordial unity had created all existence, numbers divided existence into separate entities.

A problem for this Plotinian understanding was provided by the seeming endlessness of all numbers. Because the ideal world of prototypes included in the act of self-thinking intellect contained numbers, there would have to be some perfect number containing all the existent prototypes of the ideal world. How could such a number exist if the number of numbers themselves couldn’t be pinned down? Plotinus appeared to suggest that such an imperfection concerned only souls in the material world. When we counted things, we could always find further and further numbers and no proper ending in the series. Similarly, while prototype of a line would have definite boundaries, lines in the corporeal world could apparently be extended as long as one liked.

Plotinus also noted that unlike in the intellectual level, where the numbers and quantities properly belonged, in the corporeal world things might often be numbered or quantified only accidentally. For instance, while a tree or an animal is naturally a unity, something like an army is a unity only accidentally, because the persons making up the army have no intrinsic connection to one another.

This possibility of an accidental quantification provided Plotinus with a possible answer for a question important in ancient philosophy: why do things far away from us seem smaller? Plotinus suggested as one alternative solution that from a distance we can gain information only of the essence of a visible object - in other words, its colour - while more accidental information, like the proper size of the object, are not properly transmitted to us.

Another topic, in which Plotinus noted the accidentality of quantity, was happiness. If we are happy now, this happiness is not diminished if it lasted only for a day. Indeed, even if we were happy for all eternity, the intensity of this happiness would not change in any manner. Otherwise, eternally happy entities, like stars, would never be completely happy, because they would be always becoming more and more happy.

Because quantities were part of the intellectual world, matter as such could not be quantitative, Plotinus concluded. This could even be empirically verified when a piece of papyrus was moistened by water - the whole papyrus became mixed with water, but the moist papyrus still took as much space as before. Plotinus explained this through the idea of featureless matter being formed by basic forces deriving from the intellectual level, such as forces making matter into watery or paper-like. The most central of these forces was the one involved in quantifying a piece of matter and giving it a certain volume. It was only this quantifying force that divided matter into individual bodies.

The intellectual level contained then numbers, but did it contain anything resembling corporeal entities? For instance, we know that Plotinus admitted the existence of a prototype of human beings, but would such a prototypical human sense anything? Yes, indeed, Plotinus would have answered, since even in the prototypical word of intellect one could perceive what the world is like, even if this perception was completely different from perception in corporeal world. Indeed, perception was not even the lowest activity at the intellectual level. Self-thinking intellect would think all types of beings in its self-thinking - even animals and plants - although these ideas of animals and plants would differ from corporeal animals and plants by being more prototypical and harmonious, both in themselves and in their mutual relations.

The prototypical nature of intellectual ideas requires some metaphorical light to be seen, Plotinus noted. This light, giving being to self-thinking intellect itself, derived of course from the primordial unity, which was then also the ultimate source of goodness. It was higher than mere intellect and it thus required something quite different from regular thinking to see it. Indeed, all it takes, Plotinus noted, is to immediately observe the unity, without thinking about it, but more like being swallowed by this unity. Indeed, Plotinus noted, since the primordial unity does not think, it cannot even consider things like “this is good”. Thus, one could just name this primordial unity good, reflecting the actual simplicity of primordial unity.

Science and learning in antiquity

The general tendency of ancient Greek and Roman thought has been to associate philosophy with the practice of good life and the ability to pursue such a life, although we have seen Aristotle instead relating the highest philosophy to theoretical knowledge of certain key elements of the world. Still, philosophy of the ancients was not completely separate from the questions of science, or learning in general, and for instance, early Stoics thought that full knowledge of the world and its ways was at least a necessary precondition of perfectly good life. Thus, it is justifiable to investigate Greek science and learning to shed some light on the ancient philosophy.

By far the most developed science of ancient times was mathematics. We often hear of the external form of Euclidean mathematics, which supposedly derives everything from self-evident axioms and definition through incontestable demonstrations. All of this sounds like Greek mathematics would have been quite formal and removed from ordinary experience. Yet, when we look at e.g. Euclid's Elements, we quickly find that Greek mathematics was quite heavily linked to experience. Just think about some key proofs at the beginning of the first book: Euclid asks us to move one triangle on top of another triangle and thus to conclude that the two triangles coincide. This is a primary manner in which the similar magnitude of areas is seen – through quite concrete experiment. In fact, many of the more complex proofs dealing with areas work by cutting compared areas to pieces that we could then know to be similar by relying on some previous propositions – and ultimately on direct observation. This is in fact the manner in which Euclid proves the most famous theory of elementary geometry, the Pythagorean formula.

Although Euclid called his book Elements, an average modern reader would probably find only the first book filled with recognisable questions of elementary geometry: when are triangles similar, what is the sum of the angles in a triangle, how are the angles generated by parallel line related etc.? The second book already opens up avenues unfamiliar from geometry. The book apparently deals with rectangles, but the true topic of the book are certain equations of algebra. Ancient mathematicians had as yet no way to express complex calculations in any other way than through geometric means – squaring through actual squares, multiplication through rectangles. Thus, a simple task of squaring the sum of two magnitudes was represented as a square made out of a line combined of two smaller lines – and the result was then shown to consist of two smaller squares, with sides equal to the smaller lines, together with two rectangles formed by same two lines.

The need to express such equations explains also certain uses of circle by ancient mathematicians. Circle was, of course, important because it provided a way to copy a length of one line to another place. It was also a source for more intricate tools. At the very end of the third book of Elements, Euclid proves that no matter what sort of lines you draw, from a spot outside the circle, towards the circle and cutting it in two places, the rectangle formed by the first line between the spot and the first cutting place and the second line between the two cutting places is always of the same size – and if the line merely touches the circle, the square formed by the line between the spot and the touching place equals all these rectangles. This at first somewhat uninteresting result works in many proofs as a convenient trick e.g. to express some multiples as squares.

The fourth book of Elements continues the study of circles, and its topic is especially drawing figures of certain shape within and around circles. While at first this might seem rather unimportant pursuit, fit only for decorative purposes, it is actually of utmost importance at least in two cases. Firstly, it gives a convenient manner to divide the circumference of circle to equal parts – something which is especially useful for astronomical purposes. Secondly, it can be used to estimate the area of circle. The latter case deserves a more thorough explanation.

Elements is not precisely meant to provide for exact formulas for measurement. Instead, it gives exact means for measuring relations between different geometrical objects. In fact, the fifth book contains a theory of ratios between different magnitudes – that is, it tells us when a pair of number is related exactly in the same manner as another pair and when the pair has a larger or smaller ratio than the other pair. This is convenient in case of estimating magnitude of figures that one cannot express through simple algebraic means. For instance, in the sixth book Euclid can show that two circles have the exactly same ratio as squares on their diameters. Here Euclid is helped by what is called a method of exhaustion. By drawing polygons within and around the circles, he points out that their ratios provide limits between which the ratios of the circles must fall – and because every other ratio exceeds limits given by some of these polygons, the only alternative is the one already mentioned.

Elements is then a book on two-dimensional geometry, but it is also a book on arithmetics – or at least for certain arithmetical principles. Euclid doesn't go so far as to do exact calculations, but we do know from other sources that actual counting at antiquity was rather cumbersome. Instead, in the seventh, eighth and ninth book of Elements Euclid explicates such trivialities as ”odd times even is even”, but also goes through very intricate truths about number, for instance, that there is no highest prime number.

This theory of numbers provides also a ground for Euclid's theory of rational and irrational quantities, which is developed in the tenth book. The notion of ratios provides again a starting point: we just choose some magnitude as rational and then all magnitudes of same type that have the same ratio to the rational magnitude as a number has to another number are rational, while other magnitudes of same type are irrational. Since it is especially geometric magnitudes, or lines and areas, Euclid deals with, he makes the somewhat confusing choice that even all lines with rational squares should be called rational. Going through various relations between lines and areas, Euclid provided a crude classification of all irrationals, only to conclude that his classification fails to account for all irrationals.

The final three books of Elements extend the geometrical account to three-dimensional cases. What is most intriguing is perhaps the manner in which Euclid defines the various three-dimensional shapes through a method for their construction, for instance, when a circle is thought as moving around one of its diameters, a sphere is formed. The actual propositions of the last books are mostly rather uninspiring: we are shown in a similar manner to two-dimensional cases what are the relations of e.g. pyramids and rectangular solids with same bases and same height and we see that relations of cones and cylinders can then be determined through the method of exhaustion. The final book still closes with a true high note, when Euclid shows how to inscribe all the five Platonic shapes into balls and demonstrates that the sides of these shapes must be certain irrationals.

All the things in the two-dimensional part of Elements are accomplished through circles and straight line. In later mathematics, this was made a sort of virtue, but it seems clear that at least not all ancient mathematicians did not put as strict limits to mathematics. This is shown even by Euclid's three-dimensional mathematics and its method of constructing three-dimensional shapes, but even more clearly by the ancient treatises of various types of curves, most interesting of which are possibly the conic sections, which get their names from being generated by planes cutting the cones in different angles. The use of conic section is, once again, not arbitrary, but important e.g. for expressing relations between one- and two-dimensional magnitudes.

Another interesting point in ancient mathematics is Arhcimedes' account of how to calculate various curvilinear areas and volumes. On some level, not much has changed from Euclidean account of finding relations that hold e.g. between circles. In many cases Archimedes even uses the same method of exhaustion Euclid had applied. For instance, when trying to determine the segment of a parabola, Archimedes shows that we can limit the area of the segment through sums of triangles, and by making the triangles smaller and smaller the limits become closer and closer to a certain area. But Archimedes had also another type of proof for such theories – he showed that when put to a lever the segment could be balanced with a triangle of a certain area, which was then equal to the segment. In other words, Archimedes used mechanical principles to determine an answer to a mathematical problem. As he himself testifies, this mostly just helped him to discover the areas, but he didn't accept it as a proper proof, which could be determined only through mathematical means.

Although Archimedes did use then mechanics as a tool in mathematics, more familiar is his application of mathematics in mechanics. There was considerable interest on mechanical questions even before Archimedes, as shown by a work on mechanical problems, attributed to Aristotle, but in a sense, in Archimedes the ancient tradition of mechanics culminated. Especially important in this context is the notion of a centre of gravity, that is, a place to which we can think that the whole weight of a body is concentrated. Through this notion Archimedes can prove a number of important mechanical principles, such as the lever law – the longer the stick the more weight you can lift. Note how close to practice the Archimedean practice of mechanics is – it is all about making machines, like levers and pulleys.

Even more important field of application for mathematics was the study of stars and their comings and goings. Here especially the study of circles and angles was of primary importance. Earth and the universe were pictured as a sphere within a sphere. By carefully noting the distances between places, measuring differences of shadows and the places of stars in the sky, one could get a reasonable model of the place of the Earth in relation to the universe. Then by following the changes in the positions of the stars, and especially of the strangest of these, that is, planets, and by assuming that the movements of all heavenly objects must be somehow based on circles, Ptolemy constructed a reasonably accurate model of planetary movement.

Ancient thinkers were not satisfied with mere modelling of planetary movement, but they also tried to explain this phenomenon. The basic notions were simplistically mechanical or then compared the movement of stars to movements of animal things. Whatever the case, the stars themselves and their movement was supposed to have an effect on Earthly things through their pressure on Earth's atmosphere. Such influences were also felt to be a justification for attempts to predict key events in human life.

In addition to these astrological considerations, study of stars was quite useful in making maps – one can use the positions of stars in the night sky for approximating one's position on the globe. Here ancient mathematical learning came in contact with ancient empirical knowledge. One cannot say that Greeks and Romans would not have appreciated empirical learning. Quite the contrary, as Strabo's geographical writings and Pliny's collection of all sorts of interesting factoids show.

What was lacking was such technological advances like telescope and microscope, which have allowed us to see farther and in more detail. Also, the lack of trustworthy and fast communications often made it impossible to verify the facts collected, which makes these empirical compilations into an often tantalising mix of truths and fairy tales. One can see that there was e.g. great technological expertise on various special subjects, such as metallurgy, but this expertise was dispersed to many different individuals – while one person could know everything of, say, gold and its behaviour, he might have no idea about forging iron.

While ancient culture did have empirically discovered information on many questions of nature, the role of theoretical foundation of such information was rather ambiguous, reflecting the conflicts between various philosophical schools. The complex interactions between theory and empiria and many viewpoints on these interactions can be well exemplified through the difference of ancient schools of medicine. We have methodists, who were affiliated with the Pyrrhonic school and who accepted only some rules of thumb that had appeared to work in previous occasions. We have empiricists, who accepted immediate appearances and experiences as a tool for suggesting treatment, but who denied the possibility of any knowledge on the hidden causes of various diseases. And there were various rationalist schools, who attempted to justify their procedures through some theoretical framework, although what was a correct theory of human body was a point hotly debated.

Probably the most famous of these researchers of medicine was Galen, whose position was somewhere between empirical and rational schools. Galen despised methodists, whom he thought as mere bunglers and frauds. He was quite sympathetic of empiricists and admitted that in many cases empirical observations were enough for discovering a good treatment. He endorsed the rationalist ideal of a science of medicine based on certain foundations, but only if the foundations were correct. His preferred theory of human body was quite Aristotelian: the constituents of human body were characterised by four qualities (hotness, dryness, coldness and wetness), and if some quality took a too forceful position in some part of the body, causing imbalance and disharmony, a proper treatment was to apply opposed quality.

While Galenian medicine thus offered a comprehensive account of the biological make-up of human body, rhetoric was at least advertised as a comprehensive account of the cultural side of human beings. On the superficial level, the topic of the rhetoric is much more restricted – the production of speeches, and especially of speeches meant to be used in a court setting. Thus, the central aim of ancient rhetoric was to show how to invent good arguments for one's position, how to arrange these arguments into a coherent whole, how to express this whole in a good style, how to memorise the written speech and how to present it in a compelling manner.

Yet, writers like Cicero and Quintilian were eager to suggest that rhetoric was something more, namely the lost half of philosophy. That is, they pictured a time, when philosophical study of truths and especially of truths concerning good human life and rhetorical expertise of expressing those truths were an undivided unity. A conclusion they both draw was that a good rhetorician should also be a good person and know all the intricacies of human life.

An important part in rhetoric, Quintilian suggested was then the education of a future rhetoric. As it should be evident from his wide definition of rhetoric, Quntilian plans a careful curriculum for his imaginary students who wish to learn ,not just the tricks of the trade, but also the rudiments for becoming an ideal speaker. It appears that an ideal speaker should learn as much a possible from every topic – even if her skills are to be used in the courthouse, it is a definite possibility that some case would require expertise knowledge of some special topic. In addition to specialised knowledge, the rhetorician-to-be should also learn history, since past events often play part in modern cases. A special place in the curriculum is given to literature, which at the same time teaches the students the basics of good language, but also suggests various examples of human behaviour.

Quintilian's school of rhetoric incorporates then two important pieces of humanist learning: history and literature, both of which were important fields of study in antiquity. Study of history in ancient Greece and Rome was in a sense something quite different from academic history of our time. The ancient historians had progressed not much beyond mere chronicling of events. Often their standards of criticism were quite suspect. As a case in point, we might raise the historian of philosophy, Diogenes Laërtius, whose work is at worst a rambling collection of anecdotes from the lives of philosophers, without a proper explanation of their philosophies. The main historical guide line of Diogenes is the notion of one philosopher being a student of another, which allows him to create two lines of thinkers, but fails in explaining how the ideas of earlier philosophers led to ideas of future philosophers.

As for study of different types of literature, a scholarly criticism was quite developed. As an example, we might pick out the investigation of works of Aristotle, ancient commentaries on which could have filled libraries. An important example of this tradition is Alexander of Aphrodisias, who apparently set out to explain all of Aristotle's extant works, although only a handful of his commentaries have been passed down to us. And explain he does. Following through Aristotle's account of different types of deductions, Alexander goes carefully through every reasoning and every example Aristotle uses, expanding Aristotle's sometimes rather summarised sentences.

In some cases, Alexander picks on details that Aristotle probably did not mean as important. Why did Aristotle choose to present the three different figures of syllogisms in a certain order, Alexander asks. Pretty clearly the first figure is most important, because it is the only one allowing us to prove universal and affirmative propositions, required in scientific reasoning. Alexander takes this line of thought further and suggests that second figure, leading only to negative conclusions, is second best, because it can be used in philosophical debates, meant for refuting the ideas of opponents – and the third figure is then worst, leading only to particular conclusions and thus useful only in sophistical reasoning, in which propositions applying merely for particular cases are deceptively presented as holding in general.

At times, Alexander defends Aristotle against later critics. For instance, some Stoics had claimed that Aristotle went wrong in assuming that one cannot deduce impossibilities out of possibilities, because a possible statement ”Dion is dead” implied a statement ”he is dead” (”he” referring to Dion), but since the Greek term for ”he” could be used only of living persons, the implied statement was always an impossibility. Alexander noted that, if Stoics were correct in their grammar, then the first statement simply did not imply the latter. Furthermore, he pointed out that since Stoics believed in eternal recurrence of all things, death was always only a temporary state and one could refer to a dead person with a pronoun.

Alexander also opposed Stoics on their understanding of negation. Stoics had suggested that ”Socrates is not white” is not the negation of ”Socrates is white”, because both of these presuppose that Socrates exist, while ”It is not so that Socrates is white” does not, making it the sought-out negation. Alexander pointed out that a proper name did not necessarily refer to an existing entity, like in a sentence ”My future son will be called Eric”, which made Stoic assumption doubtful.

At other times, Alexander tried to show that certain innovations of later logicians were already accounted for by Aristotle. Thus, he attempts to show that a deduction consisting of nothing else but conditionals (”if A then B, if B then C, thus, if A then C”) must still follow syllogistic rules, because the premisses must have been ultimately deduced syllogistically.

Sometimes Alexander went clearly too far in his attempt to speak for Aristotle. A good example is latter's attempt to add necessity and possibility in his logic. Aristotle's efforts become quickly a hopeless muddle, when he distinguished many different meanings of possibility or contingency and then himself confused these various meanings from time to time. Alexander's valiant effort to make sense out of it manages just to muddle the issue even further.