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[Peter Blumsom]

1. THE DIVIDED LINE

This is for those who enjoy Platonic ideas on maths that come to the surface occasionally in his greatest dialogues. There is no doubt that both he and the Academy considered an understanding of these kind of things as fundamentally important for comprehending his philosophy. It should be remembered that The Academy was by and large a mathematical institution. The idea for this post arose from a recent conversation with a friend. She remarked that Plato must have known about the Golden Ratio because he used it to section the Divided Line (Republic - 509), and it set me thinking.

Just a few words about the Divided Line; Socrates provided us with three great ideas beginning in Book 6 of Republic. Each follows fast upon the heels of the other and is informed by its predecessor. Together they make a complete philosophical statement, three in one, which has ever haunted man's philosophical imagination.

The first was the analogy between the Sun, the powerful light which allows us to understand by sight the external world; and the Good, the Form-Principle par excellence that lights up our understanding of the inner world of mind. This is followed by the Divided Line, a paradigm which further explores the first analogy, giving us a model for the whole of existence (i.e. the visible and intelligible realms) from the point of view of the witness and what is witnessed. Then, finally, we are introduced to the famous Cave Allegory, where the previous theories are presented in action, as it were; that is from the point of view of the individual soul - always Socrates main preoccupation.

Because of Plato's deep conviction that Proportion is the fundamental basis of all produced things, he is careful to emphasise that this Divided Line is itself infused with symmetry. We know this because Socrates says, "Well, suppose you have a line divided into two unequal parts and then divide the two parts again in the same ratio …" The Greek for proportion is analogia, We may take it that this term is used in a far wider context than its technical sense in mathematics; for the Ancient Greek true judgement is based on proportion, and knowing this, Aristotle says, enables a man to achieve his goal.

Let's have a look at a simplified version of the model that Plato has given us, remembering that this particular post in not concerned with the philosophy of the model but rather the way it has been proportioned.


Fig. 1


You can see that there are two types of division in the diagram: The horizontal, which shows us the 'higher' realm, (C+D), and the 'lower' realm, (A+B). But these are both subject to a vertical division that roughly corresponds to experience/thought and their objects. I've suppressed all but the essential here, for fear it will distract, but if it stirs interest in anyone they can follow the gradual unveiling of Socrates grand vision in Republic 507-521.

As to the proportionate nature of the Divided Line, Plato is saying that each of the smaller sections A/B and C/D are exact 'analogies' of the greater sections (A+B) and (C+D). He is telling us is that the relationship between knowledge/being and opinion/becoming (i.e. A+B:C+D) is similarly reflected in the smaller portions, that is, in the world of mind, between creative intelligence/eternal forms and reason/mathematics ( A:B), or its 'lower' counterpart, belief/objects and dreaming/illusion (C: D)

We are not told the nature of this proportion, for example, as to what portion is larger and what is smaller; nor are we given the values of the ratios between the divisions. Nevertheless there are those who think that it must be the golden section. I believe the reason for this is that there is something romantically unique about the golden section (which would be best discussed in a separate post) and also that it was probably known about by Plato's contemporaries. Euclid was able to give an account of its construction and although the Elements were compiled and codified about a century after Socrates it is believed that much of its contents can be traced to Plato's Academy, which of course boasted master geometers of the calibre of Theaetetus, Leodamus, Leon and others.

This is as much as I think we need to know before turning to the construction of Divided Line itself, so without further ado let's begin to study its proportions.

I'd like to simplify the diagram even more, and just view it as a geometric structure.


Fig.2


From what Socrates has explained of the structure we know that it is extremely proportionate, but what the golden section would require would be that this proportion should be continuous. Let me talk a little about this. Proportion is of two types, continuous and non-continuous. They are easy to explain. 2 and 4, that is the ratio 2:4, is proportionate to 3 and 6 (3:6) and they are both proportionate to 49 and 98 (49:98) or indeed any other pair of numbers that share the underlying ratio of 1 to 2 (1:2). We could say that 2:4 = 3:6 = 49:98. This is usually written 2:4::3:6 etc. (where (:) stands for ratio and (::) means 'is proportionate to)

Proportion therefore arises between things that share the same ratio. However, these proportions are not connected to each other except by ratio; that is, they do not share the same numbers. This is why they are called non-continuous. (The posh word is 'disjunct' – there are no points of juncture between them.)

However, in the case of 1 and 2, and 2 and 4, 2 provides just such a juncture. 1:2::2:4, and being continuous, or conjunct, a chain of similar ratios may arise each from its predecessors: 1:2::2:4::4:8::8:16 and so on.

I am proposing to look at three different ratios so that we can see how they behave, regarding continuous proportions. First lets see how 1:2 performs as a choice for the Divided Line.


Fig.3


Here we can see that because section B is the same value as section C continuous proportion is created.

A:B::CD or 1/9:2/9::2/9:4/9

And also A:B::C: D::A+B:C+D

If, however, he had indicated that there was to be a continuous proportion throughout section D would have to equal section A+B. We can see from fig. 2 that this clearly isn’t the case. D = 4/9 and A+B = 1/3 or 3/9. So A+B is deficient by 1/9 of 'continuing' the proportion. (But remember we can gather from the text Socrates does not need this requirement.)

Now let's look at another example: 2:3.


Fig.4


Apart from the actual numbers everything we said of fig.3 may be said of fig 4 save for the fact that in this case section A+B is one unit in excess of section D. (2/5 or 10/25 is 1/25 greater than 9/25).

Is there any ratio then which will provide a continuous proportion though out both the line and its parts? Well all geometricians know that there is but one, and that is the famous Golden Proportion itself. However one of the reasons that Plato might have been reluctant to use this proportion, even if he had known about it, is that it involves an irrational number – that is, it cannot be expressed as a whole number. We call this irrational phi or Φ. It is approximately 1.618… The Greeks, especially Pythagoreans, were diffident about irrationals, although, contrary to popular belief, they were well aware of them.

This then is the way that phi would have behaved in the Divided Line:


Fig.5


As is shown in fig. 5, D is now of equal size to A+B and therefore the proportion is continued from the parts of the line into the whole.

Now the interesting thing is that the two examples I have given move towards phi .by means of unit difference (one above, then one below, etc.). This has interesting connotations in ancient maths and I will talk about this method of 'zooming in' on such irrationals as phi or root 2 in my next post.

To finish, there is another reason why Plato would not have considered it important to have this continuous proportion through both the line and its parts. Looking back to the fig.1 we can see he would surely have no desire to equate the highest level of intelligence (D) with the visible world (A+B).

Congratulations to any who have reached the end of this post without tearing their hair out. I make no apology for its subject matter. Was it not said that engraved on the door of the Academy was the edict: "Let no one ignorant of geometry enter here!" I think it there is an urgent need for these Platonic number games to be understood by those who profess to love Plato, for they are actually quite serious. This kind of speculation was obviously at the heart of daily studies at the Academy, even though their importance was even misunderstood by those that followed – including, to some extent, even Aristotle. In this they certainly had all lost the Platonic plot.

Pete

P.S. If any of you are interested in following up on the golden ratio/proportion/section etc. there is the excellent 'Sacred Geometry' by Richard Lawlor. This seems to be the best of the bunch.

P.P.S. Needless to say, if any one can find any errors in the above perhaps they can respond on the forum, or, if reticent, by private email.

[Tim Addey]

Peter, I admire your enthusiasm for mathematics and its insights and look forward to getting stuck into your latest posting once the figures are inserted. However enthusiasm occasionally runs away with us, and I wonder whether you may like to reconsider your assertion, “It should be remembered that The Academy was by and large a mathematical institution.”

The academy was first and foremost a philosophical institution: mathematics was valued for the way it could be used to introduce young minds to the highest forms of thought. In the Republic, book 7 (521c), Socrates asks, “Are you willing then, that we now consider this, by what means such men shall be produced, and how one shall bring them into the light, as some are said, from Hades, to have ascended to the Gods? Why should I not be willing? replied he. This now, as it seems, is not the turning of a shell [i.e. a trivial game]; but the conversion of the soul coming from some benighted day, to the true re-ascent to real being, which we say is true philosophy.” What follows is an outline of introductory education – gymnastic, music and maths.

The mighty Proclus, begins his Commentary on the First Book of Euclid by saying, “It is necessary that the mathematical essence should neither be separated from the first nor last genera of things, nor from that which obtains a simplicity of essence; but that it should obtain a middle situation between substances destitute of parts, simple, incomposite and indivisible, and such as are subject to partition, and are terminated in manifold compositions and various divisions. For since that which subsists in its inherent reasons remains perpetually the same, is firm and durable, and cannot be confuted, it evidentaly declares it is superior to the forms existing in matter. But that power of progression which apprehends, and which besides uses the dimensions of subjects, and prepares different conclusions from different principles, gives it an order inferior to that nature which is allotted an indivisible essence, perfectly constituted in itself. Hence (as it appears to me) Plato also divides the knowledge of things which are, into first, middle, and last substances. And to indivisible natures, indeed, he attributes an intelligence, which, in a collective manner, and by a certain simple power, divides the objects of intellectual perception; so that being divested of matter, and endued with the greatest purity, it apprehends things themselves, by a certain unifying perception, and excels the other kinds of knowledge. But to divisible essences, and such as are allotted the lowest nature, and to all sensible beings, he attributes opinion, which obtains an obscure and imperfect truth. But to middle essences (and such are mathematical forms), and to things inferior to an indivisible and superior to a divisible nature, he attributes cogitation. For this, indeed, is inferior to intellect, and the supreme science dialectic; but is more perfect than opinion, and more certain and pure."

Thomas Taylor, in his introduction to the Theoretic Arithmetic of the Pythagoreans, draws attention to one of the mysterious sayings of Pythagoras, "a figure and a step, not a figure and three oboli" - Taylor suggests that this is a exhortation to use maths as a means of ascent rather than as a subservient to mere material wealth. He says, "the whole attention of those who have applied to the mathematics, has been direct to the oboli [coins], and not to the steps of ascent; and thus as their views have been grovelling, they have crept where they should have soared.

So do soar, Peter, but remember there is a greater intellectual experience beyond mathematics.

Tim

[Peter Blumsom]

I'm very gratified by the regular trickle of visitors to this topic. I hope that now John has embedded the diagrams it will begin to make more sense. This is all new work so comments pro and con are welcome. Eventually it will find its way into a longer work and it is important to me that any confusions be dealt as they arise. So please respond with questions, or even gentle criticisms in the nature of Tim's last post (which I will address as soon as possible.)

I am a musician, not a mathematician, and became interested in numbers and proportions as gradually I found out the important role they play (heeded or unheeded) in the arts - especially music, painting, sculpture and architecture. This has been written about before, but I feel we have only scratched the surface. Much more of great importance remains to be uncovered.

Pete

[Peter Blumsom]

Tim, you have changed my thinking more than once in the past; for example, I am now an avid reader of Proclus. He was once a closed book to me. In our present encounter however I think we are already close in our ideas. If I had specifically mentioned mathesis or learning, rather than mathematica I have a feeling that it would have been quite acceptable to you. But then we would not have had the pleasure of reading your well thought out response. As it is you have given me an opportunity expand on my initial thoughts. This thread will I hope show clearly the place of number in Plato's paradigm.

I did consider that statement you questioned but left it in because I believe there is a subtle difference between your 'first and foremost' and my 'by and large'. This is encapsulated in the meaning of the two Greek terms; arche – 'principle', and pragma – 'activity'. My remark was referring to pragma and yours (I suspect) to arche. I think we would both agree that those in the Academy (which, as far as I can see, did seem to be mostly composed of mathematicians of some sort) would not have spent their time in any kind of navel gazing of the sort that Aristophanes lampoons in The Clouds. Anyone who managed to pass through that portal with the fearsome prescription, would have understood, I believe, most of the activities taking place to be mathematically based, but, and here I may concur with you, mathematics modified by dialectic. We have one or two unique insights into the pragmata of the Academy from Epinomis and Aristotle's On Soul which I will investigate in subsequent posts.

Yes, it's true I do give full reign to my enthusiasms from time to time, and sometimes this lands me in the soup. I'm not sure if this is one of those occasions because as the thread unfolds I hope to make clear the position of mathematics within the Platonic spectrum. You will see that I am not talking of what you might find in the mathematical faculty of a modern university. However it might take a post or two before my vision of number becomes clear. What fired me up was something I read in Francois Lasserre's excellent The Birth of Mathematics in the Age of Plato where in a chance remark he says "Twenty five years of thought and discussion in Plato's Academy sufficed to delimit the field of mathematics in all its breadth." – p.13. There must have been quite a lot of mathematical activity going on to achieve that!

And yet, as you infer, Plato's aim would never be the mere cultivation of techne or skill. Having all these brilliant men around him, he clearly wished to propel them along the Socratic golden path from dianoia (logical thought) towards noesis (creative intelligence) Without Plato at the helm these might have remained a motley of talents rather than a philosophical movement.

I am not interested in mathematics for its own sake, nor for its ability to calculate prodigiously. Bear with me , Tim, and you, of all people, will soon 'get it'.

Pete

PS I am trying to lay my hands on a copy of Dee's Mathematical Preface to Euclid. Do you publish it?

[Tim Addey]

Pete - Many thanks for this posting: it clarifies the mathematics behind the divided line, and not being a gung-ho mathematician myself I'm always grateful to those who get down to the basics of these things.

In response to your reply to my first posting, I did think that the description of the Academy as "by and large a mathematical institution" might be defended as referring to the method of approach rather than the final aim. In which case you are probably right, at least as regards the introductory and middle states of progress through the Academy, although the descriptions involved in the divided line passage would lead one to think that as the final section is explored, dialectic and contemplation take over from mathematical hypothesis.

There are a couple of minor but linked points which are worth exploring - if only because the three great flights of philosophic description of the sun, the line and the cave are so concentrated that it is easy to miss important details.

Firstly, I would caution against thinking of the good as a form principle at least in the normal sense of Platonist usage. Plato uses "form" and "idea" quite freely in his dialogues, and I think he is expecting us to take each usage within context. The passage concerning the Good and the Sun makes it quite clear that just as the sun is not the light it sheds, nor the visible objects it illuminates, so the Good is not the intelligible light, nor the intelligible objects which are more accurately described as forms. Forms are beings in the highest sense - that is to say they are true beings or essences; but at 509b-c Socrates says, "The Good itself is not essence, but beyond essence, transcending it both in dignity and power." The Greek scholiast's comment on Glauco's response of laughter to this statement is "this laughter is through transcendency; for the Good is uncomparable with respect to all other things." We might also note that he swears by Apollo, saying, "By Apollo this is a divine transcendency indeed!" The ancient Platonists pointed out that the literal meaning of Apollo is not (the "a" bit) many (the "pollo" bit) - in other words The Good is here The One, and not one of many forms, being transcendently above forms, and a truly simple one, against which forms are complex entities.

The second point is whether we should be calling the lowest section of the divided line "illusion." An illusion has no truth, I think; but Plato is not suggesting that sense perception is outside the realm of truth - just the most obscure of the means we have of contacting reality. The final passage of book 6 has an explanation of the four sections of the divided line as regards the soul's powers which are (in Taylor's translation) intelligence, dianoia, faith, and assimilation: assimilation is that power we have to perceive truth through the similarity of lower images to true beings. I only mention this because I have heard several commentators suggest that the shadows on the cave wall in the third of Plato's great explanations in this part of the Republic are actually evil and entirely illusory, whereas Plato, I think, is only suggesting that to take the shadows as the whole of reality or the primary reality is what leads to error.

Finally, and to link the two points above, I wonder whether your third hypothesis, using Phi or the properties of the golden section, is the best for the very reason that you suggest it might be rejected. It is all too often claimed that Plato's system is basically dualist, because it postulates a world of perfect intelligible forms, and a sense world of imperfect material objects. But if we accept that the Good is the one source of all things, both intelligible and sensible, then we actually have an all-comprehending One which gives a greater meaning to materiality than it would have within a strictly dualist system. So that even the lowest shadows and images are, mystically speaking, distant offspring of the Good. And only illusory if we misunderstand what they are.

I hope this doesn't seem too critical - it's certainly not meant to be, but an attempt to make small adjustments to your very fine explanations because of misunderstandings which others have brought to the interpretation of the Republic. I'm looking forward to more postings on the line from you Pete, in order to address my general mathematical weaknesses.

Tim

PS No, we haven't published Dee on Euclid, although we have published Proclus' Commentary on him.

[Peter Blumsom]

Tim, I must apologise for my tardiness in answering your last. Things go kind of slow on the Plato Forum and that seems the best way – time to think.

Yes, eikasia is a tricky word to translate. Liddell and Scott have it as ‘image’, ‘likeness’, 'conjecture' etc. and of course ‘icon’ comes from it. So 'illusion' (Desmond Lee’s translation) is perhaps not exactly accurate. But of course, this is a complex issue and goes to the heart of Platonism. A free man re-entering the Cave would not be bound in chains. He would ‘know’ what shadows and reflections truly are for he looked directly on them when he first moved outside its portals. The shackled man looks at lawful images and erroneously takes them to be real things, and this, as you say, is the root of the illusion. But I suppose we ought to remember that Socrates' ever present aim is the care of the un-illumined soul. “Nay”. he says, “have you not observed that opinions divorced from knowledge are ugly things? The best of them are blind.” (506c). We go to Sophist for the proper discussion of this, but I suspect that to do it justice as a topic, it would need its own thread.

Now, Tim, notwithstanding this point which you raise, I think I will stick to my guns regarding phi, for even if we omit the term 'illusion' from the composite A+B, the continuous proportion would have us making an equality of the realm of pure mind and to horaton, the visible world. Although we can’t ask him, I think Plato would not be happy with this. There is no real progression from the sub-divisions to divisions in this grande analogie which would be implied in a truly continuous progression such as, say, octaves or cell partitions in a shell. Actually I think 2 to 3 would be the preferred ratio. (Do you remember the demonstration I gave at the Prometheus Trust Conference, starting with a stamp and envelope with sides of 2 and 3, which naturally expand into the Lambda numbers.) However, I am still fascinated by phi and the Golden Section, and in my next post I will be exploring it more fully, and in a way that I hope is Platonic as well as novel.

I entirely take your point regarding the Good being both source and nourisher. I wanted to make this distinction between it and the other forms by using the words par excellence, but maybe this didn’t come over. On a mathematical thread perhaps we can say, as God is to the gods, Good is to the forms.

Thank you for your remarks, Tim. No, I don’t take them as being over critical. All the time one is learning, and remember, I’m just a rock’n’roll guitarist.

Pete

[Joseph Milne]

Dear Peter & Tim,

Your discussion here has got my philosophical antenna going, so please forgive me if I deflect for a moment away from the main mathematical discussion. After all, there is a great tradition in the Dialogues to wander off now and then.

I believe this question of the difference between “the real” and “the appearing” is very important. It belongs in the same subtle realm as particulars and universals did in Medieval thought, or between Being and beings in Heidegger’s thought. These are not all the same thing, but their distinctions are very often misconceived for similar reasons.

If we say that the shadows on the cave wall are illusions, then I think we are conflating the perceiver’s opinions with the actual shadows. It is clear in the allegory that there are shadows on the cave wall, the question is about their real nature. It is clear from the analogy that these shadows are cast from the fire behind the chained observers. It is the chains which prevent the observers from knowing this, including their lack of knowing they are chained. So, given that the shadows are the only things they can see, because of their condition, they are bound to attribute full reality to the shadows. For the mind by nature holds to whatever seems true.

This is important. For Plato (and Aristotle) the mind is oriented towards truth, just as the eye is towards light or the ear towards sound. If that were not so, then the cave dwellers would not even take any notice of the shadows. And the shadows are there by the light of the fire, and the fire is there by the light of the Sun as the source of all that is. Take away the Sun and all goes, including the cave and the cave dwellers.

One often reads that Plato is a dualist (along with Plotinus), on the basis that there is “the real” eternal One transcending entirely the “unreal” temporal plurality. Thus it is said that the “Ideas” truly exist while the images of the Ideas in the temporal world do not really exist. But this is like saying that the universal “Mankind” truly exists, but Socrates does not exist, or that there is only Being but no beings. It is for this reason that Plato has been called an idealist.

But this misses the subtlety of Plato. He is countering the materialists of his time who held that only the physical objects observed by sense existed, each one separately – a sort of nominalism of his age. Plato challenges this by seeing that universal Forms are immanent in each particular entity, and so establishing a correlation between the Ideas and temporal things, and between transcendence and immanence. His concern is to direct the mind to contemplate the distinctions this presents.

What changes here is not the objects seen or known, but the manner in which they are seen and known. This surely is of vital importance. Modern philosophy has mostly become pointless because it merely tries to get at the object – the result of Descartes – while the essential question has always been about the transformation of perception and knowledge, the transformation of the observing mind or soul.

In the allegory of the cave the ascent is from the shadows and eventually to the sun, as a sequence of objects. Yet the meaning of the allegory is that it is a journey of transformed perception in the perceiving subject.

If it is to be maintained that there are illusory things, “unreal objects” which convince us they are real, then these must have another source or origin than “the Real” as such. This is the inevitable position if a dualism of reality is to be maintained. Such a dualism presents no philosophical challenge, because neither truth nor untruth are at stake in it. It is only if the Real lies present with the Appearing that truth is invoked. In this sense all that appears is there to present the Real, the temporal the eternal, the many the One.

Intermission over. Back to the maths!
Joseph

[Peter Blumsom]

No pictures in this post I’m afraid, for here I am engaged in the serious business of laying down a foundation upon which we can explore many things in future posts. If you miss out this post you might not get to taste the full flavour of what’s to come. So get your thinking caps on!

Returning for a moment to the subject discussed in the first post, that of the Divided Line as explained by Socrates in Plato’s Republic, I gave the mathematical model in terms of A and B in figure 2. In my first and second hypotheses (figs. 3 and 4) I gave these terms the values of 1 and 2, and 2 and 3, respectively. These are very interesting ratios especially regarding the Golden Section (phi), which I went on to describe in my third hypothesis. However, before looking at this I feel the need to describe the ethos or character of the Greek approach to numbers.

In the ancient way of looking at number, the unit or the monad, was considered supreme. (There is, I fully acknowledge, a difference between the unit and unity but for this post I am going to treat them as one). Although modern mathematics pays scant respect to the One, our modern concept of ordinal numbers still preserve it in a simple and natural way. We applaud the victor as he stands on the podium in first place. No one is striving for zero place, and no prizes for coming ‘minus first’ even if it were possible. In this case ‘one’ is simply the best.

For the ancients all numbers grew from and were supported by Unity; this was the philosophical approach that Socrates sought to promote in mathematical studies when in Book Seven of the Republic he called the mind ask what unity in itself is, telling us that “the study of the unit is among those that lead the mind on and turn it to the vision of reality” [524e].

Socrates continued by advocating that in the true study of numbers, aside from the calculations of merchants and shopkeepers, this unity must always be maintained. Using numbers ‘for commercial ends’, or logistike, was the most un-philosophical study of mathematics. Indeed he thought it hardly a study at all, just a tallying process. Arithmetike, or the study of arithmos, ‘the one and the many’, described a more theoretical approach. The ‘One’, in arithmetike, cannot be ‘minced up’ as he puts it. What he means is simply that, taking the case of, say, ‘one apple’, the apple part may be subdivided as many times as you like, even until it becomes a purée, but the ‘one’ itself remains untouched by such endless division. What actually happens, he continues, is that the one simply multiplies the parts of the apple “always on guard” he says “lest the one [itself] should appear to be not one but a multiplicity of parts.” [Republic 525e].

If we think this strange isn’t it exactly how our own numbers work? Is there a number that is smaller than one? No, for if we wish to look at, say two thirds of something, both 2 and 3 are multiples of 1. We use these multiples to describe a part of some thing which is less than that thing. Think about it … or not.

The only difference between the way we look at all this and the Greek approach is that they acknowledge overtly that which has slowly dulled into the mulch of our thought. All this above can be beautifully shown in diagrammatic form, but I must leave that for later posts.

Socrates held that One is the final arbiter, and he wasn’t just talking about arithmetic. In Book Four of Republic he is asked how large the ideal state should be allowed to expand to. “As long as growth is compatible with unity,” he replies, “but no further.” [423b] The aim was always to retain unity as the numbers increased. Failure to achieve this would lead first to fragmentation and finally to chaos. (This of course applies also to our own individual ‘Republics’.)

So how was this done?

To understand this we must look at the Greek idea of comparison. The mind’s ability to compare is integral to its ability to make sound rational judgements. In the normal way the mind is quite sluggish and slow to activate, but, says Socrates, it can be prompted to thought when astonished by apparent paradox. If a thing appears to be both the same and different at the same time, the principle of reason in a man is aroused to solve this problem. He talks extensively on this in Republic Book Seven, but much earlier in Book Four he makes this statement:

“So by the friction of comparison we may strike a spark which will illuminate justice for us. Then if we apply the same term to two things, one large and one small, will they not be similar in that to which the common term is applied?” [435a]

Justice is cited here as the harmonic principle by which Difference is tempered to (made compatible with) the Same.

In Timaeus Plato describes how the World Soul forms such unifying judgements:

Whenever Soul comes in contact with anything whose being is dispersed [fragmented] or indivisible [a unity] it is moved throughout and calculates similarity and difference,” [37a]

Such, Timaeus says, is true reasoning.

All the above is an attempt to prepare us for a Platonic understanding of the numbers that goes beyond their mere operations.

Numbers, for the Greeks, came in two main forms: multiples/submultiples and what we would call fractions, but as I said, the Greeks would not allow ‘the one’ to be fractured. Later we will study anthyphaireisis, which highlights the fascinating Greek approach to fractions, but here it would simply distract my line of thought, so let’s call them fractions for the time being.

Multiples are simply ‘whole numbers’, and these, because of their wholeness, were deemed the most superior kinds of numbers by the Greeks. This superiority has passed down to us by the term integer. A whole number has integrity. This integrity has survived into the modern number system. Multiples are multiples of unity. So by 3, we really mean 3/1, and so on for all integers. Of course any multiple of a multiple remains, itself, a multiple. 3/1 x 2/1 = 6/1. Multiples in this way will never ‘fragment’. There is such an agreeable nexus of insights that occur in the vicinity of number manipulation that they need a chapter of their own to do them justice, but this is a single post with hopefully a single aim, which I haven’t even touched upon yet? (Did I hear a heckler shouting ‘Get on with it, then!’ I will, dear reader I will.)

Fractions are divided into two kinds: these have become known by the Roman terms Superparticular and Superpartient and have become associated not only with musical intervals but also generally with the relation of number to number. People are often confused by these Latin terms but if we go to the Greek originals, epimoric and epimeric, all becomes clear.

Meros means ‘part’ and mere, ‘parts’. D.H.Fowler the eminent mathematical historian translates epimoric as ‘a part in addition’ which means any number or multiple along side another which is greater by a single unit. For example, 3 with 2. On the other hand epimeric refers to ‘parts in addition’. So any number alongside another which exceeds it by more than unity, or a single part, is epimeric.

What is the significance of these terms, why were the Greeks so fussed about them and why is it impossible to understand the treasury of Platonic mathematics without first understanding this significance? We certainly need to know about them if we are going to explore the Golden Section, and also later they are crucial to the understanding of Greek music theory.

Remember Socrates stricture that increase must be compatible with the maintenance of unity. Well this goes to the heart of the matter. Numbers as multiples maintain this close relation with unity because they have no remainders, and therefore may be regarded as a unity of their own. But when multiples interact the possibility of remainders (or leimmas) arise. This is where the Socratic ‘clash of comparisons’ becomes active; factionalism arises within the Republic and Justice is tested. This, by the way is how the Greeks viewed fractions, as the interaction between two multiples.

The best fractions were the epimoric. Although not directly expressing unity and wholeness in the way of the multiple, nevertheless they express unity in their difference. For example 9 to 8 which can be expressed as the multiple, 8, in addition with one eighth of itself. As the multiple is but a step from unity, the eight may immediately convert itself to unity, and the remainder becomes an aliquot eighth part of that unity. Thus in our numerical system it may also be stated as one and one eighth.

So, as the part in an epimoric ratio is always but one step, as it were, from unity, this ‘comparison’, this little mini-republic, maintains its justice. It allows for diversity but banishes the excesses of any ‘warring factions’ within it. Do you think I’m overdoing it with my analogies? I assure you I am not. This is the way the Greeks, especially the Pythagorians, enlivened and even spiritualised mathematics. We, so immersed in objects, have completely lost this dramatic and philosophical approach to number, but I shall endeavour to use it whenever appropriate.

The primary musical intervals (which I will explain anon) all express this unifying ‘justice’ in their relations. The octave at 2:1 is the finest of relations, because, just as it sounds most near to a single note, i.e. unison, numerically it is the only relation that is both multiple (2/1) and epimoric (1+1).

Now the epimeric relations are, according to the Greeks, inferior to both Multiples and epimoric relations. These were fractions which had completely abandoned unity, by having a multiple of parts between the terms. The first of these is 5 and 3, which is also one and two parts of three. But this at least is ‘the best of the worst’ because its difference is epimoric, that is, 1+2/3; 8/5 falls away even more, as 1+3/5 even possesses a remainder that is no longer epimoric.

However, as I will soon show, the Golden Section has a secret in store for us regarding these intervals, a divine path that restores unity in a most elegant way; and of all the useful things that number can reveal to us, how to maintain contact with unity is by far the most important.

Please post if you are confused by any of this!

Pete

Joseph, I'm turning to your post now. I don't think I disagree with anything you say here, but there a few points worth adding. However things, as I said, turn exceedingly slowly on this forum. Treebeard would truly feel at home here ... hrmmmm....

[Peter Blumsom]

Dear Joseph and Tim,

You both seem to be saying that illusion has no reality, and I suppose that must be true – or must it? Plato’s thought becomes very subtle in his later dialogues:

“But now that we have brought to light the existence of the false statement, it is possible that there should be imitation of things and that this condition of the mind (false judgement) should account for the existence of the art of deception.”

So says the Stranger at a culminating point towards the end of Sophist [264d]. One could say with a certain justification that this view was adopted in order to entice the Sophist ‘into his net’. One could also argue that illusion only exists if, as Tim states, we misunderstand, and, as Joseph adds, our misunderstanding only arises because of our enchained habitude and condition.

This cave, we should remember, is a 'metaphorical cave'. It’s true that without the sun both it and its fires could not exist; but as a metaphor neither would it exist except for the fact that the soul lies deceived in its chains of illusion. This whole specific situation, its raison d'etre - which is being described (in vivid detail) by Socrates - is to show an illusion; and such an illusion, stemming from the problem of the eidolon (Ficino’s idolum), must, the Stranger tells us, be taken into account.

This metaphorical cave is re-entered once the person has come to his senses, but only to free others. The illusion, still present in them, no longer holds him or her bound. What brings the free man back is simply care of soul, the hupokeimenon of all things Platonic. One can, as the Stranger proved in Sophist, have illusion without dualism. As an afterthought, this is exactly the way Sankara described the pseudo-snake superimposed on the rope in Adviata Vedanta, which also admits no duality.

Joseph intimated that something must make take note of the shadows, to which I would append, something also must have made the shadows seem like real things. These great philosophical systems needed to become very adept in dealing with illusion .

I eagerly await your comments on this.

Pete

[Peter Blumsom]

I've been asked to supply references regarding the snake and rope. There are two that I know of, both in Sankara's Vivekachudamani (Crest Jewell of Wisdom); sloka's 110 and 138.

110: Maya can be destroyed by the realisation of Brahman, the one without a second, just as the mistaken idea of a snake is removed by the discrimination of the rope.

138: One who is overpowered by ignorance mistakes a thing for what it is not: It is the absence of discrimination that causes one to mistake a snake for a rope, and great dangers overtake him when he seizes it through that wrong notion.

I often wonder what one is meant to be the illusion in that second example!

Anyway, this is an interesting work on the subject of pseudos (deception) and well worth a peruse or three. Does anyone know more references?

Pete

[Joseph Milne]

Dear Peter,

It may be we are disagreeing over words and not substance. Even so, I have gone over the cave allegory several times in different translations and none of them specifically speak of “illusion”. The explanation Socrates gives is variously translated as that which is “becoming”, or “fleeting” as distinct from that which “truly is”. Also, the distinction lies in the difference between “the visible” and “the intelligible”. It seems from this that it is an aspect of things that the cave dwellers see, and take for their reality. This aspect is but the shadow of their real existence. A shadow as such is not an illusion, but rather it is secondary to the thing that casts a shadow. Nor does the shadow vanish when the thing that casts it is seen!

This situation seems to me to be quite distinct from the snake and the rope examples. In the first of these the rope is mistaken for a snake, while in the second a snake is mistaken for a rope. These are straightforward misapprehensions of appearances. As illustrations of Maya they show that things are not what they seem to be.

But in the cave the dwellers are not mistaking one thing for another, they are seeing only what they can see because they are bound. They are seeing the shadows cast by realities but not the realities themselves. But a snake is not cast be a rope, nor a rope by a snake. The two allegories are not illustrating the same thing, even though both may be showing a truth.

So I still feel that what Socrates is illustrating here is that in the cave only a part of reality is discerned, rather than the whole – the “underground” of reality. What holds the prisoners bound is their belief that what they see is all that there is, that there is no reality beyond what they can see.

That situation is more like the materialists who reduces all things to matter, even thought or consciousness or being. They “flatten” reality down to a single plane. That is the kind of problem the one who descended back into the cave must face when he tells the dwellers what he has seen! To them his vision is incomprehensible because they see only what their bonds permit them to see. They cannot grasp that they see only a fragment of reality. But a fragment of reality is not the same as an illusory appearance. So I do not feel it is consistent with Socrates’ intention here to interpret the cave as Maya.

Where we agree is that Plato is concerned here with the transformation of the soul and its journey towards the Good. I also agree that elsewhere Plato addresses the problem of deception, both in the Sophist and the Gorgias.

Joseph

[Tim Addey]

I think the posts on this thread are sifting through the problem and bringing some interesting nuggets to light. I guess what we need to be clear about is where the deception of the shadows lies: as we can see from Joseph's post, it really lies within the human being rather than the shadows. And we have the power to rid ourselves of the deception (since the soul is self-motive, and therefore is able to change itself for the better). Whereas the shadows are just being shadows - perfectly good for the situation in the cave where artifacts are being carried passed a fire. We would be more worried if they didn't exist.

The point is important because apart from the neglect of intellect (or, in more normal terms 'spirit') which so degrades humankind and our world, we also have a problem with our carelessness with the material world. Solving the first problem at the expense of the second is in reality no solutions at all. The manifest world - the world which Zeus the Demiurge creates as described in the Timaeus - is a good and beautiful creation made by a god who himself is good, and who is looking to the perfect paradigm in the intelligible eternal.

[Peter Blumsom]

Dear Joseph

I’ve thought about your post a lot, and I don’t think I can disagree with it as a whole. I feel we are together. But there is a subtle point to be made and it’s not easy to put it into words. Firstly I am unhappy with the term ‘part of reality’. This is to do with the old adage ‘ a miss is as good as a mile’. Reality I cannot see as having parts, certainly in the discrete sense, which seems to be the way you are using it; like the monad or point, reality is not divisible.

Now this is very relevant to what you are saying. To see shadows 'as they are' necessitates you being able to see reality 'as it is'. But these souls, fallen as chronicled in Socrates’ second palinode in Phaedrus, lost their grasp of reality when they turned from the light. They see an illusion of a shadow, and they don’t even see that, they see what they think is reality. If for one instant they saw a real 'shadow', rather than a mistaken 'reality', or the real 'rock' that it flickered upon they would wake instantly.

So their predicament is hardly different to someone looking at a rope and seeing a snake. There is no 'snake', there is no 'illusion', there is only, in one case a rope and in the other a shadow, a light generating it and a wall for it to play upon. This is the reality they do not see. You see my point, don’t you Joseph. To look at something in the particular and not from the whole, is not to see it at all, but to see its image. We have moved beyond a part of reality.

Now the term ‘image’ takes us into another cave, this time, Aladdin's, at least as far as the Greek Lexicon goes. I've been having fun with Liddell & Scott. There are a whole pool of words in the Greek that relate to 'seeming'. The word I always think of as ‘illusion’ is phantasia. In Divided Line it is used with regard to 'reflections', also to 'phantoms'. ‘Images’ themselves are eikones which on the surface (as it were!) stand for 'likenesses', 'images', 'portraits', but at a deeper, more poetic level can mean 'images reflected in water' , thus connecting with phantasia through phantasmata, or images in the mind', as Socrates describes geometrical figures, or else related to the eidolon - 'nation of images'. From the eidolon proceed the endless stream of eidola, so beloved by Ficino; these are both 'phantoms' and 'reflections', and then we have skia (c.f. Sanskrit chaya) which is 'shadow', it also refers to ‘shade’ or 'dead phantom' (as in Hades!) and you can have ai tou dikaiou skia 'the shadow (illusion?) of justice'. I'll spare you the rest!

So these are all seem to have connections with seeming, which in turn can move to deception when given the added ingredient of belief (doxa) which travels down from the level above on the Divided Line. My point is, we cannot get too pedantic about the exact meaning of this terminology that Plato uses especially in a language as metaphorical as ancient Greek.

Coda
Illusion is certainly a step beyond mere shadow. But think of a mirror, That does not give you a true picture of yourself. Now think of a image in a mirror: this is the very essence of illusion (but, mark, not yet deception!). Imagine a room, full of beautiful furniture and other fixtures. Now, in your minds eye, imagine a mirror in that room. Everything else is stable, but the mirror - what is being reflected in it? Nothing, until there is an observer, there is only a surface and a potentiality. It is the very epitome of 'becoming', which again, I’m sorry Joseph, I call an illusion, but now I'm just being down right provocative!

Pete

PS By way of an afterthought, Samkhya philosophy was also materialist as opposed to the idealism of Vedanta. Maya they considered to be composed of the three gunah, which were also described as chains, sattva being the golden chain. The 'illusion of Maya' proclaimed by Sankara was to them a reality. Maya was also a goddess presiding over the gunah, as the Goddess Necessity presided over the three Fates. There are similarities in the two systems.

[Peter Blumsom]

Dear Joseph and Tim

My thoughts on this subject of illusion > deception are beginning to stir, so I will strike while they are hot.

First I'd like to expand a little on yesterday's post.

Joseph, I am puzzled when you write:

[Joseph Milne]

Dear Peter,

I feel we progress the more we probe the allegory and in a way our different angles help to bring different aspects to light. However, you have now said so much that I can only reply by taking up again the very specific question of the nature of the shadows.

You say: