Plato, Hegel and the Pythagorean tradition

 

© 2010 Aleksandar Kandić

 

Abstract: Hegel proclaimed that there are only two philosophies: the Greek, and the German. At first glance, such statement might seem arrogant and unfair to many national or supranational philosophical traditions, as well as individual achievers - but it certainly possesses some plausibility. In this paper, I will offer brief, but strong evidence not only in favor of Hegel's claim, but also in favor of superiority of ancient Greek philosophy to German idealist philosophy. In my opinion, the culminating points of these two philosophical traditions are reached within Plato's Timaeus and Hegel's Science of Logic. Both of these works are recurrently, but quite erroneously, associated with mysticism due to the complexity of ideas exposed.

I will proceed in the following manner: first, I will discuss the basic tenets of the so-called "Pythagorean" tradition, which certainly shouldn't be associated with the Pythagoreans only, and encompasses at least five millennia of scientific investigations. In his Metaphysics, Aristotle writes: "...the Pythagoreans, as they are called, devoted themselves to mathematics; they were the first to advance this study, and having been brought up in it they thought its principles were the principles of all things." (985b). And these words of Plato add up nicely: "Geometry is the knowledge of that which always is, and not of a something which at some time comes into Being and passes away." (The Republic, 527b). Why were the concepts of number and proportion of such great importance to the Pythagoreans, and how are mathematical entities related to sensible things?

Second, I will transcend the terminological (verbal) differences between the Platonic language of Timaeus and the Hegelian language of Science of Logic by using a few visual (mathematical) examples, in order to bring together the fundamental ideas of these two works. Both Plato and Hegel, when they deal with the notion of Being, end-up dealing with the notions of measure and proportion, thus committing themselves to Pythagoreanism. However, while Hegel only declares that the essence of Being consists in measure, Plato attempts to determine what these measures are(!) and he does that in various passages scattered throughout the texts of Timaeus and The Republic. In this sense, Hegelian philosophy (as well as the whole of German idealist philosophy) might be regarded as a propaedeutic to ancient Greek philosophy.

To conclude, the logical or developmental sequence does not necessarily match the chronological sequence: the philosophy of Ancients might easily be proven to surpass modern philosophy. And that is the insight which Hegel himself reaches in his "philosophy of history of philosophy". In our times, by using multidisciplinary approach to solve fundamental problems and by unifying experimental and philosophical (mathematical) disciplines, the chances to fully understand Being become greater than ever. We have an opportunity to solve "the riddle of existence" – an opportunity we shouldn't miss.

 

I Introduction

 

The huge influence which ancient Greek philosophy exercises on practically all subsequent philosophers and philosophical traditions is indisputable among scholars, researchers, even among common people (sc. non-experts on the subject). Some would go so far as to say that "the safest general characterization of European philosophical tradition is that it consists of a series of footnotes to Plato", as Alfred North Whitehead did. On the other hand, to point out these influences, to say what they consist in precisely, and to put some convincing arguments, might be problematic and opened to debate. Sometimes the German philosophers make explicit references to the Greek philosophers, and sometimes they blatantly use their ideas without naming the sources. Naturally, in some cases, such "plagiarism" is committed without intention and spontaneously. And just like the Western philosophy can be thought of as a series of comments on Plato (Aristotle, of course, being the most important among many "commentators"), so the German idealist philosophy can be thought of as a series of comments on Kant. Fichte, Schelling, Hegel... they all wished to surpass the achievements of their famous predecessor, particularly to dissect and develop Kant's concept of the so-called "categories of pure reason", as well as the general outcomes of the Critique of Pure Reason. And maybe some of them succeeded in that ambitious venture.

 

Therefore, it is no surprise that Hegel, himself being one of the most significant representatives of  German idealism, made such an arrogant yet still very advisable statement: there are only two philosophies, the Greek, and the German. Everything else is acknowledged, but could be left out. I would even dare to say – looking from today's perspective - that a great deal of written philosophy (as well as science) would not be written at all, providing that we fully understood the Greeks. But, what are Hegel's contributions to the history of philosophy (or, to be more precise, philosophy), and what gives him right to speak in that manner?

 

The key to understanding Hegelian philosophy lies within the pages of The Science of Logic. While Phenomenology of Spirit represents a strong candidate for the title of Hegel's opus magnum, it cannot surpass the Logic, because some of the most important concepts developed in Phenomenology cannot be thoroughly understood without detailed studying of the Logic (I am principally referring to the concept of indifferent/universal difference which is introduced in the section Force and Understanding in Phenomenology). The expectations of a modern/contemporary reader are almost entirely unfulfilled: Hegel's Logic is not another "scholarly" treatise on formal rules of reasoning, but a deep philosophical discussion on the notion of Being. Accordingly, the principles of his logic are to be taken as the principles of the whole of reality. Furthermore, in the first among the three books of Logic, we learn that the fundamental concepts through which the Being is comprehended are quality and quantity, as well as that these two are unified in the concepts of measure and proportion.

 

One cannot abstain from conclusion that very similar pattern of thought is present in Plato's Timaeus. This is certainly one of the most influential and most controversial texts in the history of philosophy, but I'm not going to dwell here on the origins of its contents, nor its authenticity, etc. There exists a general agreement among researchers that Plato's Timaeus exposes one of the Pythagorean theories on the origin of the Universe, and such interpretation is supported by the fact that Being is conceived in Timeaus as some kind of a system of ratios, proportions, which is determined by the numerical values of the Pythagorean scale. Although the key sections of Timaeus are very explicitly written and employ mathematical, particularly geometrical terms, the dialogue itself remains under the veil of mystery. Also, it has been, and still is, the subject of many intense and unfounded criticisms.

 

Now, after I discuss the basic tenets of the so-called "Pythagorean tradition", I am going to point out some significant similarities between Plato's and Hegel's discourse on the nature of reality. In order to clarify my claims, I will employ some visual examples (geometrical drawings, by which the gap between quality and quantity, the immediate and the indifferent, could be bridged). Both Plato and Hegel, when they deal with the notion of Being, end-up dealing with the notions of measure and proportion, thus committing themselves to Pythagoreanism. However, while Hegel only declares that the essence of Being consists in measure, Plato attempts to determine what these measures are(!) and he does that in various passages scattered throughout the texts of Timaeus and The Republic. Also, in the second book of Logic, Hegel provides a discussion on the notions of sameness and otherness, which are central to Plato's exposition in Timaeus. He introduces the concept of essential relation as well, and he doesn't take rules of inference into consideration until the third, and final book. But, herein, I will focus mostly on the results of the first book of Logic and demonstrate that, in a broader sense, Hegelian philosophy (as well as the whole of German idealist philosophy) might be regarded as a propaedeutic to ancient Greek philosophy. Hegel, obviously, doesn't live up to his potential; and neither does Kant, nor Fichte, nor Schelling - nor any of the German idealist philosophers.

 

II What does it mean to be a Pythagorean?

 

Pythagoras is somewhat a mythical figure. One can only wonder how it is possible that a person who didn't left any writings, who didn't teach publicly and who usually avoided public appearances, turns out to be such an influential thinker and practically shapes the history of Western philosophy, science, and thus, society. Some information is indisputable: Pythagoras was born around 570 B.C.E. in the island of Samos, Ionia, he spent some time abroad studying in Egypt with the priests, maybe even traveled to Persia and other countries. He was a mathematician, he engaged in both philosophical and religious studies, and he gained a relatively big number of followers with whom he took part in the political life of Croton, the Greek colony in South Italy, where he moved from Samos at the age of 40. His involvement in local government and the establishment of the Pythagorean "theocracy" (the rule of universal, divine principles), which was very strong - sometimes even brutal - provoked a rage among its political opponents who committed a mass-murder of the older Pythagoreans, probably around 500 B.C.E. This was a hit from which they have never fully recovered. However, Pythagoras' death is still considered a mystery.

 

But the Pythagorean tradition is a much, much broader term than the Greek Pythagoreanism. As already stated, Pythagoras studied mathematics with Egyptian priests, and he may have visited other countries. While the primary sources on Egyptian, Persian, or Babylonian mathematics are not so vast, and usually point to some simple, practical applications of mathematics, it is still possible to speculate that mathematics and mathematical entities possessed a deeper, ontological meaning to the members of the clerical cast, particularly if the geometrical properties of the sacred monuments in Egypt, and generally Middle East, are taken into consideration. Therefore, Pythagoras and his brotherhood might be considered a link, a very important link, in the great chain of philosophical/scientific inquiries which encompass a period of at least five millennia, beginning from astronomical observations of Babylonian and other early civilizations, and ending with contemporary microphysics and atomism. Many thinkers and researchers are, willingly or unwillingly, "Pythagoreans", just because of the paradigms they employ. Plato is a Pythagorean, Aristotle is not much less Pythagorean than some tend to think, most of the Presocratic philosophers were, in this sense, Pythagoreans, and many centuries later, Copernicus, Keppler and Galileo proudly continue the Pythagorean tradition. Modern chemists and physicists cannot escape Pythagoreanism as well (for example, popular physicist Michio Kaku tends to speak of subatomic particles as "symphonies of strings"). The subject-matter of this paper is, however, Hegel's relationship to Platonism and Pythagoreanism, being that he proved himself as one of the greatest Pythagoreans ever.

 

Now, before we get to the key sections of Plato's and Hegel's works, we shall turn to the often quoted but not thorough enough analyzed Aristotle's account on the Pythagoreans and their teaching, which, in my opinion, is safe to accept, being that Aristotle had a very good overview of the Presocratic philosophy:

 

...the so-called Pythagoreans, who were the first to take up mathematics; not only advanced this study, but also having been brought up in it they thought its principles were the principles of all things. (Metaphysics, 985b, translated by Ross, W. D, cursive A. K.)

 

This passage clearly indicates that the Pythagoreans had a different conception of mathematics than we have today. In modern times, "mathematics" is usually understood as a formal system which does not reflect reality, least the things (unless it is somehow applied to the purpose of processing empirical data and solving practical problems). This view on mathematics is accurately expressed by the following words of Albert Einstein: "...as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality"[1]. But, would the fathers of the mathematical discipline be willing to approve of such a view? Are we missing something? These words of Plato complement the view which Aristotle ascribed to the Pythagoreans, and they give us the proper direction in understanding the main purpose of mathematics:

 

Geometry is the knowledge of that which always is, and not of a something which at some time comes into Being and passes away. (The Republic, 527b, translated by Shorey, P.)

 

Let it be said that Pythagoras and his followers coined themselves the word "mathematics", which literally means "study", "learning" - but not just any learning (in the sense of remembering facts and information). It is systematic, thorough learning, which is accompanied by understanding. Out of this reason, the Pythagoreans were divided into mathematicians (Greek: mathematikoi), who possessed true, genuine knowledge, and acousmatics (Greek: akousmatikoi), or "listeners", who only transmitted information without proper understanding.

 

What beliefs, then, could be attributed to the upholders of the Pythagorean doctrine[2]?

 

(1) Contrary to the modern conception of mathematics, there should be certain unchanging, immovable and timeless aspects of physical reality which constitute the proper object of mathematical studies, and thus, receive the name of “mathematical entities”. If such entities didn’t exist in reality, we wouldn’t be able to conceive them in our minds. Furthermore, their relationship to sensible things is that of cause to effect;

 

(2) Mathematical entities are not mere constructions of the human mind. Instead, they are abstracted from experience. For example, mathematical entities could be some elemental, simple things which combine into sensible, physical objects;

 

(3) By abstraction, by observation and by constant reexamination of sensory data, the mind discovers mathematical entities as the first principles of reality. Then, it is able to deduce all the facts about world beginning from these first principles, which have to be some kind of patterns encompassing all possibilities. That makes them universal laws.

 

This brings us to the most important question... If the principles of mathematics are the principles of all things, and if mathematical entities constitute physical reality, in which manner, then, are the mathematical entities related to physical, sensory objects, and how are we supposed to envision them? How would a true Pythagorean answer this?

 

Actually, most of the Pythagorean metaphysics and natural philosophy is not aprioristic, as it might seem at the first glance; instead, it is derived from empirical observations. It is generally recognized that the Pythagoreans studied music and musical intervals, and that, consequently, the mathematical and philosophical disciplines were conceived. But they were not interested in aesthetic dimensions of music, at least not as much as musicians. By experimenting with monochord, a single string instrument, Pythagoras discovered that certain mathematical ratios correspond to certain musical intervals. For example, a ratio of 1:2 (a division of the whole string in half) produces an octave. Any string vibrating in the 1:2 ratio produces an octave - no matter how long it is, or from which material it is made. And this vibration is perceived by our sense of hearing. This lead Pythagoras to a significant idea which will lay foundations not only for mathematical physics, but for all natural sciences as well: the sensory qualities are always determined by quantitative relations. Number and proportion became the language of creation. David Fideler, in his Introduction to the Pythagorean Sourcebook and Library writes: "...but for Pythagoreans Number is a universal principle, as real as light (electromagnetism) or sound. As modern physics has demonstrated, it is precisely the numeric, vibrational frequency of electromagnetic energy – the "wavelength" – which determines its particular manifestation. Pythagoras, of course, had already determined this in the case of sound."[3]

 

A string on the monochord can be divided in different ratios which produce different musical intervals. While some of these intervals are euphonic, coherent, symmetrical, some are cacophonic, incoherent. Musical harmony and - as we shall see - Existence, are therefore envisioned as an indefinitely complex system of proportions. In his paper Lestvična deoba po zlatnom preseku, Predrag Milosavljević published the arithmetical and geometrical principles by which the values of the so-called "big intervals" of the Pythagorean scale are obtained[4].

 

(1) The whole tone } 8:9 (=0.888...), or inversely 9:8 (=1.125). Five whole tones and two semitones comprise an octave.

(8 / 9)5 / (256 / 243)2 = 1 / 2;

 

(2) The fourth } 3:4 (=0.75), or inversely 4:3 (=1.333...). Comprised of two whole tones and one semitone.

 

(8 / 9)2 / (256 / 243) = 4 / 3;

 

(3) The fifth } 2:3 (=0.666...), or inversely 3:2 (=1.5). Higher than the fourth for a whole tone.

 

(3 / 4) · ( 8 / 9) = 2 / 3;

 

(4) The octave } 1:2 (=0.5), or inversely 2:1 (=2). The octave encompasses the fourth and the fifth.

 

(3 / 4) · (2 / 3) = 1 / 2.

 

The most important thing I want you to realize now, is that the number in itself cannot have an empirical function. Only geometry has the power to bridge the gap between quantity, and the perceived quality. As Fideler puts it: arithmetic studies the number in itself, geometry the number in space.[5] According to Milosavljević, the geometry of the "big intervals" of the Pythagorean scale is simple, and can be represented by the interplay of halves (or radiuses):

 

veliki intervali

 

Picture 1. The geometry of the Pythagorean scale

 

Notice the following intervals on the drawing, and their corresponding numerical values:

 

AB = BC = BD = 1 (the unit);

AO = BO = 1:2 (the octave);

BS = 8:9 (the whole tone);

BK = 3:4 (the "perfect" fourth);

BJ = 2:3 (the "perfect" fifth).

 

We have now successfully set-up the ambient in which we are going to read and, hopefully, understand the most important paragraphs of Plato's and Hegel's works.

 

III Plato versus Hegel

 

In this section, I will briefly demonstrate that Plato and Hegel, when they deal with the notion of Being, employ the completely same pattern of thought. Hegel's philosophical language is overcomplicated and often labeled "mystical", or "exotic" (particularly when it comes to the passages of Phenomenology of Spirit in which he speaks of inverted world). He doesn't offer a lot of facts, but a compilation of thoughts and ideas which are remarkably eloquently expressed. His writing style is, actually, closer to that of Hermann Hesse, or some other Nobel prize laureate in literature, than to philosophical, which should be overwhelmed with facts and arguments. Still, it is not too difficult to distinguish the philosophical elements from monumental writing and literary figures.

 

Although confronting Plato's and Hegel's fundamental ideas is a tremendous task and surely requires a book to complete, I have managed to select a few passages – mostly from The Science of Logic and Timeaus, the works which I have already designated as the "culminating points" of the German idealist and the ancient Greek philosophical traditions – that easily put both philosophers into the "Pythagorean" context. Furthermore, the ideas which Hegel presents as his own are so blatantly Pythagorean-Platonic that one could accuse him of theft without any hesitation. Let's have a look at the following passage:

 

Measure is now determined as a correlation of measures which constitute the quality of distinct self-subsistent somethings — or things. The relations of measure just considered concern abstract qualities like space and time; those now about to be considered are exemplified in specific gravity and later on in chemical properties, i.e. in determinations characteristic of material existence. Space and time are also moments of such measures but their relationship no longer depends simply on their own nature because they are now subordinated to further determinations. Among the determining moments in sound, for example, there is the time in which a number of vibrations occur, and the spatial element of length and thickness of the vibrating body; but the magnitudes of those ideal moments are determined externally. Space and time are no longer in a relation of powers but in the ordinary direct relation, harmony being reduced to a quite external simplicity of numbers whose relations can be grasped with the utmost ease and hence afford a satisfaction falling entirely within the element of sensation since there is an absence for spirit of figurate conception, fantasy, abstract thought, and the like. In that the sides which now constitute the measure relation are themselves measures, but at the same time real somethings, their measures are, in the first place, immediate measures and as regards their relations, direct relations. It is the inter-relationship of such relations that is now to be considered in its progressive determination. (Nauka Logike, Book I, pp.334-5, translated by A. V. Miller, bold letters A. K.)

 

This citation comes from the section of The Logic entitled Real Measure, which is found near the end of the first book. The very beginning of the passage indicates that the quality of things - or "self-subsistent somethings", as Hegel puts it - is to be thought of as a correlation of measures (previously, I used the terms "system of proportions" and "interplay of halves" to denote this concept). Such idea was never explicitly formulated in Kant's writings, and possibly, it wasn't even hinted... Later on, Hegel expresses his intention to comprehend physical phenomena, such as gravity, chemical properties, etc. through the paradigm which we have clarified at the end of the second section of this paper (see!), and then he provides an example similar to the one by which Fideler has tried to explain the relationship between the Pythagorean theory of harmony and findings of modern physics. While Hegel's insight that in immediate experience measure, or the correlation of measures, is not distinct from things (the measure being the thing itself) has to be correct, he still fails to comprehend the manner by which real measure can be represented abstractly (ideally), without loosing relationship to its reality, its particularity. He jumps to conclusion that the real measure is somehow restricted to sensation, but the fact is that it comprises both ideal and real, spiritual and physical. The Pythagorean geometry successfully solves this problem: as we have demonstrated by Picture 1, when realized/actualized in the audible range, the system of ideal ratios of the Pythagorean scale generates the phenomena which are perceptible by our senses.

 

On the other hand, Plato does not spend tens of thousands of words just to establish the paradigm by which our investigations will be guided. His writing is not only propaedeutic - he actually attempts to provide some definite anwers. And that is something which deserves a lot of respect, even if he isn't successful in his efforts. But, day by day, contemporary science proves the Pythagorean-Platonic theories. Of course, Plato does provide a great deal of preparation to his readers in the form of discussions scattered throughout the dialogues, just to get us to The Republic and Timeaus which are the only ones that don't end aporetically, and which expose his own findings on the nature of reality. One of the most significant passages in Plato is the one from Timaeus, where he explains the structure of the so-called "world soul" (the living essence of the Universe) by the intervals of the Pythagorean scale:

 

Now as regards the Soul, although we are essaying to describe it after the body, God did not likewise plan it to be younger than the body; for, when uniting them, He would not have permitted the elder to be ruled by the younger; but as for us men, even as we ourselves partake largely of the accidental and casual, so also do our words. God, however, constructed Soul to be older than Body and prior in birth and excellence, since she was to be the mistress and ruler and it the ruled; and, He made her of the materials and in the fashion which I shall now describe. Midway between the Being which is indivisible and remains always the same and the Being which is transient and divisible in bodies, He blended a third form of Being compounded out of the twain, that is to say, out of the Same and the Other; and in like manner He compounded it midway between that one of them which is indivisible and that one which is divisible in bodies. And He took the three of them, and blent them all together into one form, by forcing the Other into union with the Same, in spite of its being naturally difficult to mix. And when with the aid of Being He had mixed them, and had made of them one out of three, straightway He began to distribute the whole thereof into so many portions as was meet; and each portion was a mixture of the Same, of the Other, and of Being. And He began making the division thus: First He took one portion from the whole (1); then He took a portion double of this (2); then a third portion, half as much again as the second portion, that is, three times as much as the first (3); the fourth portion He took was twice as much as the second (4); the fifth three times as much as the third (9); the sixth eight times as much as the first (8); and the seventh twenty-seven times as much as the first (27). After that He went on to fill up the intervals in the series of the powers of 2 and the intervals in the series of powers of 3 in the following manner: He cut off yet further portions from the original mixture, and set them in between the portions above rehearsed, so as to place two Means in each interval, one a Mean which exceeded its Extremes and was by them exceeded by the same proportional part or fraction of each of the Extremes respectively; the other a Mean which exceeded one Extreme by the same number or integer as it was exceeded by its other Extreme. And whereas the insertion of these links formed fresh intervals in the former intervals, that is to say, intervals of 3:2 and 4:3 and 9:8, He went on to fill up the 4:3 intervals with 9:8 intervals. This still left over in each case a fraction, which is represented by the terms of the numerical ratio 256:243. And thus the mixture, from which He had been cutting these portions off, was now all spent. (Timaeus, 34b-36b, translated by Lamb, W. R. M, bold letters A. K.)

 

The numerical values and ratios which Plato cites here can be systematically obtained from the geometrical drawing discovered by Milosavljević, and presented in Picture 1. Plato was inspired by the idea that the very same ratios detected in the audible range play significant role in the structuring of matter, both at the microscopic level, where we deal with atoms and molecules, and at the macroscopic level, where we deal with stars, planets, etc. Also, it is noticeable that by introducing the three forms of Being: indivisible (unchangeable), divisible (subjected to change), and the "midway" one which is a mixture of the previous two, Plato attempts to solve one of the greatest puzzles of German idealism, the relationship between "the ideal", and "the real". Their unity is understood in terms of the geometry of the Pythagorean scale.

 

There is, however, another fine example of Plato's supremacy over Hegel, and consequently, over the other German idealists. After he made a quick transition to the concept of essence at the very end of the first book of Logic, throughout the second book Hegel clarifies that by "essence" he doesn't mean some kind of "transcendent reality", but the law (the truth) of Appearance (sc. simple, immediate perception). This truth is the essential relation:

 

The truth of Appearance is the essential relation, the content of which has immediate self-subsistence; simply affirmative immediacy, and reflected immediacy or self-identical reflection. At the same time, it is in this self-subsistence a relative content, only and solely as reflection into its other, or as unity of the relation with its other. In this unity the self-subsistent content is a posited, sublated content; but it is just this unity which constitutes its essentiality and self-subsistence; this reflection into other is reflection into itself. The relation has sides because it is reflection into an other; thus it contains within itself its own difference, and the sides are a self-dependent subsistence, since in their mutually indifferent diversity they are disrupted within themselves, so that the subsistence of either side equally has its meaning only in relation to the other or in their negative unity. (Nauka Logike, Book II, p.123, translated by A. V. Miller, bold letters A. K.)

 

What is this mystical "essential relation", and how are we supposed to envision it? Did Plato, perhaps, have an analogous concept, and did he provide a thorough description, in terms which could be rendered mathematically? We have to get back to Timaeus:

 

But it is not possible that two things alone should be conjoined without a third; for there must needs be some intermediary bond to connect the two. And the fairest of bonds is that which most perfectly unites into one both itself and the things which it binds together; and to effect this in the fairest manner is the natural property of proportion. For whenever the middle term of any three numbers, cubic or square, is such that as the first term is to it, so is it to the last term, and again, conversely, as the last term is to the middle, so is the middle to the first, then the middle term becomes in turn the first and the last, while the first and last become in turn middle terms, and the necessary consequence will be that all the terms are interchangeable, and being interchangeable they all form a unity. (Timaeus, 31b-32a, translated by Lamb, W. R. M, bold letters A. K.)

 

Plato refers to a specific proportion, a specific relation, and he considers it "essential" in the sense which is very similar to Hegelian. The only relation which possesses the properties Plato enumerated (the most important being that the whole relates to the bigger part as the bigger part to the smaller, and vice versa) is the so-called "Golden mean". Its formulation might be given in the following manner:

 

AB : AE = AE : (AB – AE)

 

If AB = 1, then AE equals the irrational number 1 / Φ = 0.618... which has many unusual characteristics[6]. Any understanding of the Golden mean cannot be thorough and complete without geometrical representation, and thus Milosavljević published a simple construction which he called "primary"[7]:

 

zlatni presek


Picture 2. The primary construction of Golden mean

 

Plato's definition of Golden mean, which is not only formal but ontological as well, is rather unknown among modern mathematicians and philosophers. Among professional researchers, there exists relatively big animosity towards Golden mean for which the main culprits are numerous 20th century  "mystics" and pseudo-philosophers. They managed to distort the genuine meaning of Golden mean and its empirical function. While it is true that Golden mean plays an important role in the structuring of matter, this role is only constitutive: it is the fundamental principle (or the "essential relation") by which other magnitudes, quantums, ratios, etc. are generated and comprehended. It always resides in the domain of ideal, and cannot be immediately measured. (Compare Picture 2 to Picture 1.)

 

So, one of the most important lessons of the first book of The Science of Logic is the following:

 

Abstractly expressed, in measure quality and quantity are united. (Nauka Logike, Book I, p.315, translated by A. V. Miller)

 

And one of the most important conclusions Fideler gave on the subject of Pythagoreanism:

 

...the truth of the matter is that it is precisely through the Pythagorean approach (the concept of measure, A.K.) that quantity (number) and quality are discovered to be integrally related. (The Pythagorean Sourcebook and Library, p.48)

 

For the final, unquestionable proof that quantity and quality are united in measure, I will turn to the contemporary experimental results which refer to the structure of matter, and not the "aesthetic" dimensions of the musical intervals. In the last, seventh section of the paper Lestvična deoba po zlatnom preseku, which I have already mentioned several times, Milosavljević claims that the water molecule possesses a simple geometry, a simple logic by which it is structured, and that its geometry is actually related to the geometries of the Pythagorean scale and Golden mean. Let's have a look at the drawing, and the corresponding numerical values[8]:

 

molekul vode


Picture 3. The geometry of water molecule

 

Instead of using the letters "A" and "D", I have marked the appropriate intersections by the letter "H", which denotes the two hydrogen atoms (the letter "O" was already convenient to denote the oxygen atom). The angle <HOH = 2 · <HOC = 104,47751219...° is the precise, ideal angle (or measure) which determines the sensory qualities of water molecule in gaseous state. This angle is the essence of water. Bear in mind, though, that the drawn lines constitute only a part of an indefinitely complex system of proportions, and that any geometrical drawing, or natural structure, possesses an indefinitely big number of structural levels ("fractality" is the best term to describe this property of Nature[9]). And for more information on water, visit the webpage of Martin Chaplin (http://www1.lsbu.ac.uk/water/). Beside providing the biggest on-line resource of experimental data and literature on water, Chaplin makes a very profound statement on the introductory page of his Internet project: "Water is the most studied material on Earth but it is remarkable to find that the science behind its behavior and function are so poorly understood (or even ignored), not only by people in general, but also by scientists working with it everyday." Hopefully, the results published by Milosavljević shall bring an improvement to this frustrating situation. (Compare Picture 3 to Picture 2 and Picture 1.)

 

If we possess no rational, psychological, or empirical reasons to reject the presented facts, and I believe we don't, then we are safe to say that the Pythagoreans, and their most famous "representatives" in the philosophical branch Plato, and Hegel (as well as other less known and less influential thinkers who followed the same path), truly did capture the essence of Being. Existence is explicable, measureable, and governed by logical, universal principles which can be formulated in terms of geometry and arithmetic, proportion and number. Taken in this sense, the principles of logic truly are the principles of reality.

 

Could this be the fulfillment of Hegel's dream?

 

IV Conclusions

 

When I got the idea for this reaserch, my intention was not to write a scholarly treatise on the subject (anyway, that wouldn't even be possible as it would require much more space). My intention was to offer an integrative, original and intellectually provocative outlook by quickly interconnecting the fundamental ideas and notions of a few great philosophers and philosophical traditions. Although I am fully aware that the text is demanding and difficult to comprehend, I still hope that the exposed material will encourage the reader to study the source texts himself in detail, and thus extend his understanding of the Pythagorean philosophy, Platonism, Hegelianism, as well as the whole of Existence. To that purpose, the extensive list of literature is given below. This relatively short text might be considered an introduction to a bigger study which I'm now thinking to develop from its key points.

 

From the standpoint of contemporary natural sciences, the approach to the study of Being which the Pythagoreans exercised does not lack any theoretical rigorosity, exactness, nor modernity at all. The only obstacle which the ancient Pythagoreans couldn't overcome, was the lack of technical equipment and experimentation by which they could verify some of their more ambitious theories on the origin of the Universe and the structuring of matter. Still, being that they had so much trust in "the power of the mind" (sc. cognitive capabilities), such obstacles did not prevent them to extrapolate their findings on musical harmony to other spheres of natural philosophy, and thus pave the way to development of modern experimental sciences.

 

But, how do Plato, Hegel, and other German idealists fit into that picture? While the general constatation that the German philosophers demonstrated a lot of courage in dealing with the deepest philosophical problems cannot be brought into question, there still remains an impression, particularly after reading a paper such as this one, that they didn't offer so much as the ancient Greek philosophers did. As I claimed in the introductory section, the Germans just don't live up to their potential: the highest reaches of their philosophical systems are still "critiques", "outlines", "introductions" - very lengthy introductions - and that goes especially for the "young and green" F. W. J. Schelling, who managed to publish not only The Outline of a System of the Philosophy of Nature, but also an Introduction to the Outline of a System of the Philosophy of Nature. Among the German idealists, Hegel demonstrated the most strength. He is less abstract than his colleagues. His Pythagorean-Platonic approach to the study of Being leaves him just a step away from the big answers we all seek, as philosophers, or as ordinary humans.

 

Ironically or not, one of the most interesting concepts Hegel implied by his "philosophy of history of philosophy" is that, somehow, the chronological sequence does not necessarily match the logical, or developmental sequence, which means that the predecessors might surpass their successors (he called the philosophy of Parmenides "the first genuine philosophy" which is, if taken in strictly chronological sense, untrue). I have briefly demonstrated that this is precisely the case with Hegelian and Platonic philosophy, and respectively, German idealist and ancient Greek philosophy. For, whenever Hegel has a verbal concept, a subjective discourse on Existence by which he expresses his vague feelings and intuitions, Plato has an objective, visual concept which can be formulated in terms of mathematics and judged by experimental results. It this sense, the German idealist philosophy is to be regarded as a propaedeutic to ancient Greek philosophy, which offers much more complete answers.

 

Clearly, the philosophical problem of Being demands a multidisciplinary approach. The flowering of experimental sciences, and integration of numerous scientific and philosophical disciplines, from mathematics to physics, metaphysics, chemistry, ethics, aesthetics, biology, anthropology, cosmology, etc. might easily place us in a "privileged" position of being able to solve the riddle of existence. It is up to us if we are going to take advantage of the situation, and evolve higher than ever before. And certainly, the Pythagoreans, Plato, and many other ancient Greek philosophers, will be our best guides to that path. The confusion into which the German idealists have fallen, without even being aware of it, is accurately expressed by the following words of Schelling:

 

The regularity displayed in all the movements of Nature - for example, the sublime geometry which is exercised in the motions of the heavenly bodies - is not explained by saying that Nature is the most perfect geometry. Rather conversely, it is explained by saying that the most perfect geometry is the productive power in Nature; a mode of explanation whereby the real itself is transported into the ideal world, and those motions are changed into intuitions which take place only in ourselves, and to which nothing outside of us corresponds. (First Outline of a System of the Philosophy of Nature, pp.193-194, translated by Peterson, K. R.)

 

Had he found out what this geometry precisely is and how it operates, he would have known that Nature is perfect geometry, and that the real already is ideal.

 

SELECTED BIBLIOGRAPHY

 

[Primary sources]

 

Aristotel: Metafizika; Beograd: Kultura, 1971, pp.3-67 and pp.285-366

Aristotle: Metaphysics in Barnes, Jonathan (ed.): The Complete Works of Aristotle (ed.); Princeton, New Jersey: Princeton Univeristy Press, 1984, pp.1552-1572 and pp.1688-1728

Dekart, Rene: Praktična i jasna pravila rukovođenja duhom u istraživanju istine in Dekart, Rene: Rasprava o metodi; Valjevo: PŽM, 1999, pp.61-77 and pp.99-118

Diels, Hermann: Predsokratovci, Sv. 1; Zagreb: Naprijed, 1983, pp.92-115 and pp.349-364

Fihte, J. G: Učenje o nauci; Beograd: BIGZ, 1976, pp.11-50 and pp.112-157

Hegel, G. W. F: Fenomenologija duha; Beograd: Dereta, 2005, pp.5-86 and pp.353-362

Hegel, G. W. F: Nauka Logike, vol. 1; Beograd: BIGZ, 1987, pp.46-153 and pp.181-367

Hegel, G. W. F: Nauka Logike, vol. 2; Beograd: BIGZ, 1987, pp.5-59 and pp.94-139

Hegel, G. W. F: Phenomenology of Spirit; Oxford: Oxford University Press, 1977, pp.1-103 and pp.479-494

Kant, Imanuel: Kritika čistog uma; Beograd: Dereta, 2003, pp.33-197 and pp.303-307

Plato: The Republic in Hamilton, Edith and Cairns, Huntington (ed.): The Collected Dialogues of Plato; New Jersey: Princeton University Press, Princeton, 1989, pp.688-844

Plato: Timaeus in Hamilton, Edith and Cairns, Huntington (ed.): The Collected Dialogues of Plato; New Jersey: Princeton University Press, Princeton, 1989, pp.1151-1211

Platon: Država; Beograd: BIGZ, 1993, pp.136-326

Platon: Timaj; Beograd: Mladost, 1981, pp.51-139

Schelling, F. W. J: First Outline of a System of the Philosophy of Nature; Albany: State University of New York, 2004, pp.193-232

Schelling, F. W. J: System of Transcendental Idealism; Charlottesville: University Press of Virginia, 2004, pp.1-154 and pp.215-236

 

[Secondary sources]

 

Annas, Julia: An Introduction to Plato's Republic; New York: Oxford University Press, 1981, pp.190-293

Annas, Julia: Aristotle’s Metaphysics, Books M and N; New York: Oxford University Press, 1976, pp.1-77

Barker, Stefan: Filozofija matematike; Beograd: Nolit, 1973, pp.36-64 and pp.166-188

Beiser, Frederick (ed.): The Cambridge companion to Hegel; Cambridge: Cambridge University Press, 1993, pp.86-210

Copleston, Frederick: A History of Philosophy, vol. 7; London: Search Press, 1963, pp.32-58, pp.105-125, pp.159-203, pp.241-244 and pp.261-276

Ghyka, Matila: The Geometry of Art and Life; New York: Dover Publications, Inc, 1977, pp.ix-xii and pp.1-174

Gika, Matila: Filozofija i mistika broja; Novi Sad: Književna zajednica Novog Sada, 1987, pp.3-52 and pp.200-248

Guthrie, K. S. and Fideler, D. R. (ed.): The Pythagorean Sourcebook and Library; Grand Rapids, Michigan: Phanes Press, 1987, pp.19-54

Guthrie, W. K. C: History of Greek Philosophy, vol. 1; Cambridge: Cambridge University Press, 1962, pp.146-340

Guthrie, W. K. C: History of Greek Philosophy, vol. 4; Cambridge: Cambridge University Press, 1975, pp.434-561

Guthrie, W. K. C: History of Greek Philosophy, vol. 5; Cambridge: Cambridge University Press, 1978, pp.241-320

Houlgate, Stephen: The Opening of Hegel's Logic; West Lafayette, Indiana: Purdue University Press, 2006, pp.9-28 and pp.115-143

Lawlor, Robert: Sacred Geometry (Philosophy & Pracrtice); London: Thames & Hudson Ltd, 1982, pp.4-109

Livio, Mario: The Golden Ratio; New York: Broadway Books, 2002, pp.1-91 and pp.229-254

Mandelbrot, Benoit: The Fractal Geometry of Nature; New York: W. H. Freeman and Company, 1977, pp.1-24

Marić, Ilija: Platon i moderna fizika; Nikšić: Društvo filosofa i sociologa Crne Gore, 1997, pp.11-128 and pp.257-264

Milosavljević, Predrag: Lestvična deoba po zlatnom preseku in Phlogiston, vol. 15; Beograd: Muzej nauke i tehnike, 2007, pp.5-72

Paton, H. J: Kant's metaphysic of experience, vol. 2; London: George Allen and Unwin, 1970, pp.17-80 and pp.439-462

Pavlović, Branko: Filozofija prirode; Beograd: Plato, 2005, pp.5-127

Pavlović, Branko: Tajne dijaloga Timaj in Platon: Timaj; Beograd: Mladost, 1981, pp.5-49

Russell, Walter: The Universal One, vol. 1; Swannanoa: University of Science and Philosophy, 1974, pp.158-165

Stern, Robert: Hegel and the Phenomenology of Spirit; New York: Routledge, 2002, pp.43-70

 

[Electronic sources]

 

Chaplin, Martin: Water Structure and Science; http://www1.lsbu.ac.uk/water/ (20.06.2010.)

Einstein, Albert: Sidelights on Relativity; http://www.gutenberg.org/dirs/etext05/slrtv10.txt (20.06.2010.)

Hegel, G. W. F: Science of Logic; http://www.marxists.org/reference/archive/hegel/works/hl/ (20.06.2010.)

Kandić, Aleksandar: Dekartov projekat univerzalne matematike; http://frag005.com/dekartov-projekat.htm (20.06.2010.)

Kandić, Aleksandar: Mathematics as a Science of Reality: Pyhtagoras, Plato, Descartes; http://frag005.com/mathematics-athens.htm (20.06.2010.)

Kandić, Aleksandar: Mathematics: Modern and Ancient; http://frag005.com/mathematics-seoul.htm (20.06.2010.)

Kandić, Aleksandar: Reinventing Philosophy; http://frag005.com/reinventing-philosophy.htm (20.06.2010.)

http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/phi.html (20.06.2010.)

http://en.wikipedia.org/wiki/Fractal (20.06.2010.)

http://en.wikipedia.org/wiki/Pythagorean_tuning (20.06.2010.)

http://en.wikipedia.org/wiki/Science_of_Logic (20.06.2010.)

http://en.wikipedia.org/wiki/Timaeus_(dialogue) (20.06.2010.)


[1] Einstein, Albert : Sidelights on Relativity (http://www.gutenberg.org/dirs/etext05/slrtv10.txt)

[2] See: Kandić, Aleksandar: Mathematics: Modern and Ancient (http://frag005.com/mathematics-seoul.htm)

[3] Guthrie, K. S. and Fideler, D. R. (ed.): The Pythagorean Sourcebook and Library, p.21

[4] See: Phlogiston, vol. 15, pp.28-31 and Diels, H: Predsokratovci, Sv. 1, p.359. While the "big intervals" constitute the symmetrical base of the scale, the "small intervals" constitute the necessary asymmetrical complement.

[5] Guthrie, K. S. and Fideler, D. R. (ed.): op. cit, p.34

[7] Milosavljević, Predrag: Lestvična deoba po zlatnom preseku in Phlogiston, vol. 15, p. 8

[8] See: Phlogiston, vol. 15, pp.54-63

[9] See: Mandelbrot, Benoit: The Fractal Geometry of Nature, pp.1-24