Scott A. OlsenPO Box 3612Ocala, Florida 34478 USA
Lesser = 1 / f . This is, to my knowledge, a novel hypothesis, and my goal in this paper is to argue its potential for validity. Recall the following facts: Given these facts and the hypothesis above, we have:
The fact that the ratio of the Greater to the Lesser is f
Plato, primarily as a proponent of Pythagorean philosophical doctrines,[2] was very careful with what he did and did not reveal, being under an apparently severe oath of secrecy. Both his writings and lectures are enigmatic, and he only very carefully and subtly provides the clues with which the observer may be capable of uncovering the inner doctrines for themselves. His method in the written dialogues appears to be similar to his reported approach in the Academy, where he would propose the problem to be solved. He would present the problem, puzzle, anomaly, apparent contradiction or incomplete result, intending that the attentive student would abduct an explanatory hypothesis. Thus, the underlying intention was to get the observer (Academy member or dialogue reader) to abduct (hypothesize) an appropriate solution or answer, rather than to accept the dead end, apparent contradiction or incomplete result. There are several Platonic puzzles and unsolved issues. Some
of these arise within the dialogues and others in remarks made
by Aristotle and early commentators regarding Plato's doctrines.
When several of the key puzzles are viewed in conjunction, they
help point in the direction of the required solution. In particular,
I will argue that the Aristotle makes it eminently clear that within the Academy,
Plato professed Two Principles, principles that were involved
in the construction of the Forms (Universals or Archetypal Numbers),
as well as the Sensibles (Particulars) of our Empirical World.
The First Principle is generally acknowledged. It is the Good
of Plato's Although there are important references to this Second Principle
in the dialogues (especially the
And as these Two Principles were ontologically prior to and
causally involved in the manifestation of both the Forms and
Sensible things, it should not be surprising that Plato held
them to be of the utmost importance. Thus we learn from Aristotle's
pupil and commentator, Alexander, that these Two Principles were
"more important than the Ideas" ( Now according to Aristotle and others, what Plato presented to members of the Academy and in public lectures was not always identical to the content of the written dialogues. We learn from Simplicius that:
Thus, there is considerable evidence of Plato avowedly professing that there are the Two Principles of the One and the Indefinite Dyad. The great mystery has always been, what exactly does Plato mean by the Indefinite Dyad, or as he called it, Excess and Deficiency, or the Greater and the Lesser. Aristotle does tell us:
And of course all Sensible objects of this world are derivative from these original Principles via the Forms or Numbers. Now in the
As Keith Critchlow indicates:
Plato gives us the Ö2 triangle for the construction of the Cube, and the Ö3 triangle for the construction of the Tetrahedron, Octahedron and Icosahedron. But the triangle (or the root numbers embedded in it) necessary for the construction of the Dodecahedron is most conspicuously absent. Regarding the Ö2 and Ö3 primitive triangles, however, Plato states cryptically (and yet very revealingly for the astute student):
The missing triangle for the construction of the Dodecahedron
must involve (purely from mathematical considerations) the Golden
Section. But why is the Golden Section to be so protected within
the Pythagorean tradition? I would like to propose here that
for Plato it is its In the
Now following the Pythagoreans, Plato places a great deal
of emphasis on numbers, ratio (
In the - indicating a kind of ontological proportion linking together his worlds of Being and Becoming, and
- providing an epistemological framework for attaining deeper insights into the nature of reality.
He does this with the Sun Analogy (
Usually when commentators attempt to determine how the line
is to be divided, they fail to first fully consider the underlying
significance of Plato asking the reader to divide the line unevenly.
Now it seems clear to me that if Plato is concerned primarily
about continuous geometric proportion, as he appears to assert
in : longer segment :: longer
segment : shorter segment.Plato then asks us to cut each of those two segments again in the same ratios, namely Golden Cuts. In effect what Plato is asking us to do is to perpetuate the continuous geometric proportion into the four subdivisions of his Divided Line. What he has effectively done through a series of Golden Cuts is to bind his so-called Intelligible and Sensible Worlds (and their subdivisions) together through continuous geometric proportion employing the Golden Section. Kenneth M. Sayre of the University of Notre Dame, in his 1983 book, posed a very interesting challenge to anyone who would propose a Golden Section solution to Plato's Divided Line. I will call it the "Sayre Challenge." He writes:
I accept the challenge. In fact it will assist us in explicating the underlying significance of the Indefinite Dyad and the One. Let us take a Pentagram, which inherently contains numerous Golden Cuts, and extract one of its lines while retaining its points of intersection. Thus, we have line Next take a pair of compasses and, rotating line segment One consequence of this construction is what some have referred
to as the "anomaly" that line segments
I want to suggest that, to the contrary, the abductive solution to this so-called anomaly helps lead to the most fruitful insights. Plato knew exactly what he was doing. He was very subtly embedding the Indefinite Dyad into his Divided Line, expressing it through continuous geometric proportion. I propose that the inner two line segments Thus, D In the
This agrees with my hypothesized associations. The Greater
and the Lesser are to be related not only to one another, Greater
: Lesser (a single proportion exhibiting the f We can now verify the "Sayre Challenge": (here we have used the facts that 1/f
+ 1 = f, 1 + f
= f The Divided Line presents Plato's Two Principles, both the
One and the Indefinite Dyad. And its accomplishment is that it
effectively counters the "worst difficulty" argument
of
In other words, the whole number 2 can be generated from the Indefinite dyad. Indeed, recalling that Greater - Lesser = 1, we have Greater + Unity - Lesser = 2. Now the real secret of Plato's Indefinite Dyad (in addition to generating the whole numbers) is that it may be employed to derive the other crucial roots (Ö2 and Ö3) necessary for the construction of four of the five Platonic solids. The following construction is the result of carefully combining two insights, one that I had regarding Ö3, and one that Mark Reynolds shared with me regarding Ö2. While contemplating the nature of the Indefinite Dyad, which I had already decided must be the Golden Section and its reciprocal, I had a dream in which I saw the Greater and the Lesser as the legs of a right-angled triangle of which the hypotenuse was Ö3. I jumped out of bed and grabbed a pair of compasses, straightedge and pencil. I did the construction and lo and behold it was true (as we'll verify mathematically below). Then a year later I had the good fortune of meeting Mark Reynolds. I shared this construction with him, and he in turn showed me how Ö2 was derivable, in a very similar manner, from the Lesser and the square root of the Greater as legs of a right-angled triangle.[7] (In both cases I was attending the KAIROS Summer School studying under Dr. Keith Critchlow and John Michell - Buckfast Abbey in Devon, England in 1997, and Crestone, Colorado in 1998.) The construction of the Indefinite Dyad Template follows.
Though the construction has already proven to harbor many wonderful
properties, notice in particular the Quadrilateral - Construct Square
*ABCD*with side*AB*= 1. - Construct the Golden Rectangle
*ABGH*from Square*ABCD*using diagonal*FC*. - Construct Square
*AIJH*by extending line*AB*to*I*, and line*HG*to*J*(in both cases extend the line lengths by the equivalent of*DH*or 1/f), and connect*I*to*J*. *AH*=*AI*=*IJ*=*HJ*= f.- Extend line
*DC*intersecting line*IJ*at*K*. *DK*= f.*DH*=*KJ*=*CK*= 1/f.- Cut line
*AH*at*L*by rotating side*GB*(at point*G*). Using the Pythagorean Theorem on right triangle*GHL*, since*GL*= f,*GH*= 1, and 1 + f = f^{2}, we obtain*HL*= Öf. - Cut line
*HJ*at*M*by rotating line*HL*(at point*H*);*HM*= Öf. - Using the Pythagorean Theorem,
*DH*^{2}+*HM*^{2}=*DM*^{2}; thus we have,(1/f) ^{2}+ (Öf)^{2}= *DM*^{2}1/f ^{2}+ f= *DM*^{2}(1 - 1/f) + (1/f + 1) = *DM*^{2}2 = *DM*^{2}*DM*= Ö2 (here we have used the facts that 1/f^{2}+ 1/f=1, and 1/f +1=f).[9] - Using the Pythagorean Theorem,
*DH*^{2}+*DK*^{2}=*KH*^{2}; thus we have,(1/f) ^{2}+ (f)^{2}= *KH*^{2}1/f ^{2}+ f^{2}= *KH*^{2}(1 - 1/f) + (1 + f) = *KH*^{2}(1 - 1/f) + (1 + 1/f + 1) = *KH*^{2}3 = *KH*^{2}
so that*KH*= Ö3 (here we have used the facts that 1/f^{2}+ 1/f=1, and 1/f +1=f, and 1 + f = f^{2}).
(For completeness, we note that Hence, Ö2 and Ö3
are ultimately derivable from the Greater (
[2] I have argued this point exhaustively elsewhere,
and would at this stage simply refer the reader to my dissertation
[Olsen 1983], especially pp. 45-121. [3] It should be noted that there is definite evidence
that Plato and his pupils were working with the Golden Section
in the Academy. Heath relates, "We are told by Proclus that
Eudoxus 'greatly added to the number of the theorems which Plato
originated regarding the section, and employed in them the method
of analysis [abduction].' It is obvious that the section was
some particular section which by the time of Plato had assumed
great importance; and the one section of which this can safely
be said is that which was called the 'golden section,' namely
the division of a straight line in extreme and mean ratio which
appears in Eucl. II. 11 and is therefore most probably Pythagorean"[1956:
v. 1, 137]. [4] Others have proposed the principle of the golden
section in the Divided Line (see [Brumbaugh 1954] and [Des Jardins
1976]) or a Fibonacci approximation to it [Dreher 1990]. But
these prior attempts do not embody the essential features of
the Greater and Lesser (Indefinite Dyad) in relation to the One,
as I depict them below. For a critique of the sufficiency of
earlier arguments (taken individually) that attempt to establish
the golden sectioning of the Divided Line, see [Balashov 1994].
[5] We could also simply begin with the line as Unity,
and then do the subsequent cuts. The initial golden cut of the
line would then give us a longer and a shorter segment, namely,
1/ f
and 1/f [6] Here I am thinking of the Implicate Order that David
Bohm, the physicist, suggested is enfolded into the outer Explicate
Order of our world. This "patterned order" enfolded
into Nature appears to be closely related to the continuous geometric
proportion of the One and Indefinite Dyad as expressed, for example,
in Fibonacci and Lucas whole number approximations in minerals,
plants, animals, microtubules and DNA. See, for example, [Bohm
1980; Dixon 1992; Goodwin 1994; Penrose 1994; Martineau 1995].
[7] Neither Mark nor I labor under any false illusion
that we have discovered these things; we have simply rediscovered
them independently. Perhaps we are uncovering what for many in
the past may have been restricted or esoteric knowledge. [8] Plato's ontology is based upon his Pythagorean belief
that the Divine manifests throughout our world through the Numbers.
Thus, he appears to be suggesting that the Principles of the
Numbers, namely the Indefinite Dyad in relation to the One, generate
or unfold the Divine within all things through this number matrix.
During this construction, I would ask the reader to keep in mind
a very important comment by Johannes Kepler: "Geometry has
two great treasures: one is the theorem of Pythagoras; the other
the division of a line into extreme and mean ratio [golden cut].
The first we may compare to a measure of gold; the second we
may name a precious jewel" [quoted in Hambidge 1920]. This
construction in effect embodies the application of these "two
great treasures". [9] This is Mark Reynolds's very important contribution
that allows this "Platonic Template" to work. [10] Notice also the relationship between the sides of
the three squares,
Ancient Philosophy 14:
283-295.Barnes, Jonathan, ed. 1984. Bohm, David. 1980. Brumbaugh, Robert. 1954. Cairns, Huntington and Edith Hamilton, eds.
1971. Condat, Jean-Bernard, ed. 1988. Critchlow, Keith. 1963. Critchlow, Keith. 1994. The Platonic Tradition
on the Nature of Proportion. Pp. 133-168 in Des Jardins, G. 1976. How to Divide the Divided
Line. Dixon, Robert. 1992. Green Spirals. Pp. 353-368
in Dreher, John Paul. 1990. The Driving Ratio
in Plato's Divided Line. Dunlap, Richard A. 1997. Fideler, David. 1993. Goodwin, Brian. 1994. Hackforth, R., trans. 1972. Hambidge, Jay. 1920. Heath, Thomas L. 1956. Herz-Fischler, Roger. 1987. Kirk, G.S. and R.E. Raven. 1975. Koshy, Thomas. 2001. Kramer, H.J. 1990. Plato and the Foundations of Metaphysics. J.R. Caton, ed. and trans. New York: State University of New York Press. Kuhn, Thomas. 1970. Martineau, John. 1995. Michell, John. 1988. Glenn R. Morrow. trans. 1992. Mueller, Ian. 1992. Mathematical method and
philosophical truth. Pp. 170-199 in Olsen, Scott A. 1983. Olsen, Scott A. 2002. Plato, Proclus and Peirce:
Abduction and the Foundations of the Logic of Discovery. Pp.
85-99 in Pacioli, Luca. 1982. Peirce, Charles S. 1931-1958. Penrose, Roger. 1994. Shadows of the Mind. Oxford: Oxford University Press. Sayre, Kenneth M. 1983. Schwaller de Lubicz, R.A. 1998. Stewart, Ian. 1998. Taylor, A.E. 1926. Forms and Numbers: A Study
in Platonic Metaphysics (I). Taylor, A.E. 1928. Taylor, Thomas. 1804. Taylor, Thomas. 1983. Thomas, Ivor. ed. 1957. Thompson, D'Arcy Wentworth. 1928. Excess and
Defect: Or the Little More and the Little Less. Thompson, D'Arcy Wentworth. 1992.
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