Query: The Quality of Number |
**ORIGINAL QUERY:** Date:
Wednesday, 27 November 2002 11:27:42 +0100
From: Kim Williams
<kwilliams@kimwilliamsbooks.com>
In the Nexus 2002 round table
discussion, Robert Tavernor said that in his *Ten Books on
Architecture* Leon Battista Alberti writes about both the
**quantity** and the **quality** of numbers. To quote Tavernor,
"Thus, [Alberti] talks about the importance of measuring
buildings, of the *experience* of measuring buildings, so
that there is that one-to-one relation with things: that numbers
are not just abstract things, they describe qualities too. And
he particularly talks about the quality of number in a universal
sense, in terms of its relationship to ourselves and the meaning
of number beyond ourselves. So I think it's very difficult to
teach mathematics to architects today without also emphasising
the quality of number. Understanding these qualities comes only
through experience." **Can anyone explain exactly what
might be meant by the "quality" of number?**
*NNJ* READERS'
RESPONSES: From: Robert
Tavernor <absrwt@bath.ac.uk>
What I meant by the distinction between quality and quantity
is set out in two pieces of writing:
R. Tavernor, *On
Alberti and the Art of Building*, Yale University Press,
1998: esp.
chapters 5-8,
and more recently,
R. Tavernor, "Contemplating Perfection through Piero's
Eyes", in *Body
and Building*, George Dodds and Robert Tavernor, eds.,
MIT Press, 2002: chapter 5.
------------------------------------------------- From: James
McQuillan <jasmcq@yahoo.com>
There are several puzzlements about this discussion of mathematics
and quality, e. g., why did Prof. Tavernor not explain what was
at stake in the phrase from Alberti, and why the questioner might
have demanded enlightenment? The major puzzlement, however, arises
out of the subsequent commentary where no one has pointed out
that our understanding of mathematics has profoundly changed
due to the scientific revolution, when figure and number, the
mathematicals, were severed from all invisible meaning whatsoever,
giving rise to our disenchanted world view (Weber). Indeed the
very moment of this sundering can be accurately pinpointed to
Galileo's doctrine of relating mathematics to physics resulting
in the new physico-mathematics (classical physics). Hitherto
physics was a dialectical investigation under Scholastic and
other modes, where discourse guaranteed truth, and was not mathematical
at the highest level, figure and number being abstractions from
mundane corporeity mediating with eternal realities.
While Pythagoreanism/Platonism favoured mathematics as a prominent
key to transcendental reality, figure and number were never cut
off from the fullest participation in all other forms such as
eternity and the virtues. But Galileo now designated the mathematicals
as primary qualities, rendering all other qualities as secondary,
and thus setting in chain the deep confusion that pertains until
today. The success of classical physics undermines the weight
of traditional mathematics that was not instrumental but analogous
and metaphoric, and not used to investigate but to contemplate
nature. Astronomy is the obvious overlap, but remember that this
activity was the contemplation of superlunary elements, whose
movements had to be reconciled with perfect forms, as Plato and
so many others later demanded.
Many contributors to the discussion have hailed the doctrine
of Nicolas of Gerasa as celebrating the presence of quality in
the mathematicals, to which I would add the *Theology
of Arithmetic* attributed to Iamblicus (3rd c., A. D.).
Finally on the Galilean doctrine that lies at the basis of scientific
method, there is no clearer exposition of arguably the greatest
intellectual rupture that Western civilisation has ever experienced,
than the magisterial statement of E. A. Burtt's *Metaphysical
Foundations of Modern Science*, which has always been
reprinted since its issue early in the last century.
-------------------------------------------------
From: Lionel March <lmarch@ucla.edu>
**YESTERDAY**
When Alberti was writing, the words ‘quantity’ and
‘quality’ still retained their Aristotelian roots.
In relation to ‘number’ they carried the specific meanings
derived from Nicomachus and his Latin translator Boethius.
Where Nicomachus (*Introduction to Arithmetic*, II.21.5,
24.1, 25.5) writes poiòthV,
Boethius translates *quantitas*; and for (II.21.5, 24.1)
posóthV, the Latin *qualitas*.
Nicomachus (II. 23.4) gives an example of each:
[The arithmetic] proportion, therefore, partakes in equal
quantity in its differences, but of unequal quality; for this
reason it is arithmetic. If on the contrary it partook of similar
quality, but not quantity, it would be geometric instead of arithmetic.
Thus, 2, 4, 6 shows equal differences of 4 - 2 = 2, and 6
- 4 = 2, but different ratios between terms, 4 : 2 = 2 : 1, but
6 : 4 = 3 : 2. While 2, 4, 8 gives different differences 4 -
2 = 2 and 8 - 4 = 4, but the same ratios, 4 : 2 = 2 : 1 and 8
: 4 = 2 : 1. A ‘difference’ is quantitative, a ‘ratio’
is qualitative. The harmonic proportion is said to be neither,
but is ‘relative’ (II.25.5). Nicomachus is forcing
the three most established proportions into three of Aristotle’s
ten categories — quantity, quality and relative. Alberti
does not fall for this, although he acknowledged Nicomachus’s
arithmetical authority.
Hans-Karl Lücke finds few uses of *quantitas* and
*qualitas* in *De re aedificatoria*. The passages in
which these words occur do not suggest that Alberti had any precise
concern for the quantitative and qualitative aspects of number.
That issue derives from recent critical interpretations of his
writing and his practice. In any event, no interpretation would
fit the excessively narrow and forced meaning to be found in
Nicomachus via Boethius.
In *Categories*, Aristotle’s initial examples of
quantity are ‘two cubits long’ or ‘three cubits
long’; and of quality, ‘white’, ‘grammatical’.
Later, Aristotle considers both discrete and continuous quantities
— multitudes such as natural numbers are discrete; magnitudes
such as lines, surfaces and solids are continuous. Aristotle
admits, as a type of quality, ‘figure and shape’, ‘straightness
and curvedness’. Thus, from an Aristotelian perspective,
in giving shape to an architectural work, Alberti is engaged
in qualitative decisions, but in dimensioning the work he is
acting quantitatively.
A pediment is qualitatively ‘triangular’, but its
dimensions are quantitatively 24 feet long to 5 feet high. Now,
if someone were to say that the pediment was Pythagorean, a relative
statement would have been made since the triangle in the pediment
relates to the 5-12-13 Pythagorean triangle.
For relations of number to many other matters in the Renaissance,
see my *Architectonics
of Humanism: Essays on Number in Architecture*, 1999.
**TODAY**
These former arguments are embedded in the intellectual frame
of the Italian fifteenth century. Coming to our own age thought
has changed radically. The Aristotelian model no longer applies.
Starting with the re-emergence of Platonism at the very beginnings
of the ‘scientific revolution’ with Nicholas Cusanus
in Alberti’s own time, to Kant, to Hegel, to Peirce, to
Frege and Russell, Husserl, Wittgenstein and on, the ‘categories’
have tumbled into disarray and obsolescence, and with them any
meaningful meaning of ‘quantity’ and ‘quality’,
let alone ‘number’. By example, according to my contemporary
at Cambridge, John Horton Conway, the concept ‘number’
may now be understood as subordinate to the concept ‘game’.
I suggest, a contemporary approach would be computational
with respect to ‘number’ and semiotic with respect
to reference and usage. As in a Stiny shape grammar, it might
still distinguish between ’number’ and ‘shape’,
between the defining elements of shape — point, line, plane
— and shapes themselves, but certainly not for the categorical
reasons given by Aristotle.
**TOMORROW**
I have no interest in teaching architects mathematics. I use
the contemporary language of mathematics, when convenient, to
describe formal, spatial occurences in architecture. The architecture
comes first, the mathematics is secondary. Proportion, symmetry
and arrangement may call upon the language and concepts to be
found in elementary computational theory, combinatorial theory,
and topology. At most, the student’s attention might be
drawn to the fact that such material exists and that it may have
relevance in future architectural work. Period.
In giving an example of the number 64, I might present architectural
expressions such as these in which each design is made from 64
unit cubes. Across the center is a line of length 64. Below it
are rectangular planes of area 64, 2 x 32, 4 x 16, and the 8
x 8 square. At bottom left is a triangular arrangement based
on the generation of square numbers from the sum of odd numbers,
8 x 8 = 1 + 3 + 5 +7 + 9 + 11 + 13 + 15. Next to this, the truncated
triangle is based upon the generation of the cube numbers from
subsets of the odd numbers, 4 x 4 x 4 = 13 + 15 + 17 + 19. Below
the plane areas are solids. In the first diagonal are cuboids,
2 x 2 x 16, 2 x 4 x 8, and, top, the cube 4 x 4 x 4. In the next
diagonal, some pin-wheel designs, and at bottom right, threedimensional
versions of the planar, triangular designs to the left. Above
the center diagonal line are courtyard and lightwell schemes.
Whereas the mathematical question might be ‘compute the
floor area of a scheme’, the architectural design question
is ‘find a scheme, or schemes, that have a given floor area’.
The mathematical question, in such cases, is expected to have
just one, unique answer — correct, or incorrect. The architectural
question has no particular answer, each architect will give an
answer bearing her, or his, own distinctive signature —
no longer a normative matter of right, or wrong, but of preference
both ethical and aesthetic.
------------------------------------------------- From:
Emanuel Jannasch
<ejannasch@hfx.eastlink.ca>
The query concerning the quality of number is provocative
and timely. If the posted replies are any indication our age
is not well equipped to provide an answer. We all seem to avoid
the issue and revert to familiar discussions of dimension and
quantity. It seems to me that the quality of numbers is altogether
unrelated to dimension. It has more to do with the positive integers
as embodied in groups of things. Here are four examples or four
aspects of what might be considered numerical quality:
- Greek mathematicians spoke of numbers as having shape: certain
numbers (4,6,9,16...) are said to be square, because groups of
respective size could be arrayed in a square matrix; whereas
other numbers (3,6,10,15...) are triangular, for analogous reasons.
- Both Hebrew and Greek numerology, if I'm not mistaken, considered
numbers as having a sex. Odd numbers are male (arranged in a
line they have a central member) and even numbers as female (they
have a central space).
- The fundamental difference between evenness and oddness
in the matter of collonades, column grids, naves and aisles,
etc., is second nature to architects. In a more complex case,
Palladio said that the piers of bridges ought to be even in number,
because Nature has given animals legs in even numbers, because
it avoids the problems of building in mid channel, (and leaves
it free, presumably, for shipping) and, summing up, because "this
compartment is more agreeable to be looked at." (Bk III
chap X p 2)
- In the Poetics of Architecture Tzonis and Lefaivre discuss
the pervasiveness of the number three in classical architecture,
with reference to the Aristotelian division of texts into beginning,
middle, and end. The column is divided into analogous parts:
the principal shaft with the capital and a base at each end acting
as boundary elements. (The column can be read upwards or downwards,
as construction or load path.) In a rectilinear plan the classical
trisection is applied in depth as well as width, leading to the
prototypical ninesquare arrangenment which differentiates corners
and sides as well as center. The Poetics of Architecture is one
of the few modern works that takes much of an interest in the
architectural quality of number.
We could easily add other situationds to which characteristic
numbers of elements apply, or other circumstances in which characteristic
numbers arise.
The character of integers gets diminished as they get larger
and their differences get relatively smaller, but the small numbers
have such distinct and powerful character as to inspire mystical
awe. Unity, duality, trinity, perpendicularity... literary and
religious meanings of a number derive from its structural character,
not the other way around. It is not hard to see how devotees
of these integral aspects of number considered irrational numbers
to be lesser things, even illicit or sacriligeous. But this concrete,
embodied understanding of numbers is mathematically primitive.
It takes us back to grade school, to the counting and adding
of apples. We can begin to understand our contemporary disinterest.
Tweaking dimensions may seem like the more sophisticated application
of mathematics, but to my way of thinking it is a secondary operation.
And I would say that particularly in architecture - where bounding
and separating elements have substance and thickness - crisp
mathematical approaches to proportion are seldom as satisfying
as they set out to be. Perhaps they are essentially graphic than
architectural pursuits. The quality of numbers, on the other
hand, understood as the arrangability of specific numbers of
elements, is a fundamentally architectural quality. I would go
so far as to call this character the architecture or the tectonics
of number.
------------------------------------------------- From:
Carroll W. Westfall
<Carroll.W.Westfall.2@nd.edu>
Numbers have meaning. They not only relate to one another
within a system of numbers (1+2=3) but they also point to things
outside themselves. These are their qualities. Thus, 1 is unity
and God and the unity of all things in God. The quality of 2
is man, both body and soul, or Christ’s two natures. Thus
Alberti’s successor and near contemporary Filarete makes
a city that has a double-square plan (two squares set at 45 degrees
to one another) for man to live in. The quality of 3 is the trinity,
and salvation, or “on the third day”. It also encompasses
God (1) and man (2) in Christ (God + Man=3). The quality of 4
is the evangelists, and seasons of the year that God made, the
trials of the last way with the 4 horsemen of the apocalypse,
etc. The quality of five, I forget. Six is the days of creation,
7 the cycle of days and the Sabbath, 8 is salvation (7 plus 1,
or the eternal day after the seven days of life, and the 7 ages
of man). You get the idea.
In the world Alberti lived in, in the world everyone lived
in before the Enlightenment, numbers had meaning, and that meaning
provided their quality. When a person saw something that was
clearly 3-fold (e.g., the façade of Sant’Andrea,
with its three bays, the larger arched one in the center opening
to the church), those qualities came to mind.
And there is this, which I’ll mention but not explain.
In Greek, there are no numbers. Alpha is one, beta is two, etc.
This means that Greek words can be converted easily to numbers.
There are certain numbers that are fundamental in a theological
sense (144 for example) that turn up in certain words. As I said,
I do not know this material well, but this suggests is richness.
And finally, there is that wonderful book by George Hersey,
*Pythagorean
Palaces: Magic and Architecture in the Italian Renaissance*,
Cornell University Press, 1976, which discusses how numbers relate
to one another in meaningful ways within meaningful systems that
generate the proportions of buildings. This is an important but
quite neglected topic.
A further, final point: these are natural symbols, not conventional
ones, i.e., they are in nature (when nature is understood in
a pre-Enlightenment sense), not in custom.
------------------------------------------------- From:
George Hersey <Glherse@aol.com>
I go into this question in considerable depth in my book *Pythagorean
Palaces: Magic and Architecture in the Italian Renaissance *(1976).
-------------------------------------------------
From: Matthew
Landrus <matthew.landrus@wolfson.oxford.ac.uk>
This may not be exactly what you are looking for, but it seems
to me that Alberti's 'quality of number' refers to the estimated
number, as opposed to the exact number. For Leonardo and other
fifteenth century artist/engineers, this quality refers to the
geometrical process of estimation. Of course, exact numbers are
called quantita discontinua, because they are discontinuous,
and continuous numbers were known geometrically as quantita continua.
I interpret this latter quantity as the quality discussed by
Alberti, Leonardo and Pacioli. For Alberti - who wrote for a
new audience of previous aristocrats looking for a proper income
- 'quality of number' refers to a link between the goldsmith's
trade and the liberal art of mathematics. Though the goldsmith's
guild of sculptors and painters may not have had formal training
in the abacus schools, they used mathematical procedures involving
first the estimation of measuring and then the task of exact
measuring. One geometrical example: root 1 and root 2 solutions
were used, instead of the golden section, as practical design
solutions for irrational number proportions. Leonardo refers
to a 'pyramidal law' around 1505, which is a geometrical and
flexible quality of numbers that explain Medieval principles
in statics, dynamics and mathematics. These principles include
impetus theory, optical theories, and the rule of three equation.
Thus, I've found some evidence that Alberti's quality of number
refers to the non-exact continuous quantities of geometry during
the fifteenth century.
------------------------------------------------- From:
Jonathon Giebeler <giebeler@infinito.it>
Quality of number is the relation between the abstract thought
of number and its physical dimension. I believe this is what
Tavernore discusses when he says, "understanding these qualities
comes only through experience." How does one respond to
a column that is 90 feet high and one that is 9 feet high? The
columns may share the same style, purpose, and proportion, but
the effect - the human response - is quite different. The space
that the columns define and their relationship to the viewer
have a different "quality" altogether.
The interesting thing is that if I presented the elevations
of both columns but scaled the larger down by 10, you would not
be able to tell the difference between 9 and 90.
How do I feel as I stand next to a column, what about the
dimensions of the stairs as I climb and descend, or the height
of a passage way to another room, or that of the entrance? All
of these are questions of quality of number or scale. To design
with precise proportion and style requires a certain amount of
knowledge, but to translate this design into something that creates
a meaningful human experience requires a certain amount of understanding.
And understanding comes only from experience. It cannot be acquired
intellectually.
To understand the quality that number creates one must relate
the thought of number on paper to the reality number in physical
space - one must measure and experience, not view from a distance.
:-)
------------------------------------------------- From:
Chandler Davis <davis@math.toronto.edu>
Of course it was not long after Alberti that philosophers
from Bishop Berkeley to Goethe agreed that mathematics, and quantitative
physical theories as well, systematically ignored the qualities
of things. Prior to the Renaissance, it was common for philosophers
to ascribe qualities to numbers in a mystical way (3 is perfect,
for example), but that's not what Tavernor is talking about and
so probably not what Alberti talked about either. (If I measure
a building as 4 m, or (close enough) as 13 ft, which are relevant
to the building, the number-theoretical properties of 4, or of
13?)
------------------------------------------------- From:
Linda Wellner <alphagirl51@yahoo.com>
Check the book by Michael Schneider - "*A
Beginner's Guide to Constructing the Universe *"
. Also, you can go to Michael's
website for quick info on number quality.
Also, see Keith Critchlow's book, *Islamic
Patterns: An Analytical and Cosmological Approach.*
------------------------------------------------- From:
Rachel Fletcher <rfletch@bcn.net>
A very good introduction to this complex subject is Thomas
Taylor's classic, *The
Theoretic Arithmetic of the Pythagoreans*.
------------------------------------------------- From:
Berndt Wegner <wegner@math.TU-Berlin.DE>
The nearest solution I know is what has been described in
the book *From the Golden Mean to Chaos* by Vera
de Spinadel. But I shall make further checks.
------------------------------------------------- From:
George W. Hart <george@georgehart.com>
Just a guess, but I suspect he may be referring to what we
call dimensions or units. Undoubtedly his students would confuse
and mis-convert between different units of length, just as today's
students do.
------------------------------------------------- From:
Judith Newmark <jnarch@earthlink.net>
in olden times (Greek/pre-Greek) numbers were used to represent
universal qualities such as concordance. The ratios especially
were thought to represent the universal sphere of "god"
as evidenced in the quote: "as above, so below" meaning
that things in the universal sphere were comprehensible and knowable
through numbers.
The golden ratio or golden section is the best example, used
by Pythagoras, phi (1:1.618) describes sunflowers, snail shells,
human proportions and Leonardo da Vinci used it to draw the universal
man.(Circle in a square).
Matilla Ghyka's book *The
Geometry of Art and Life*, and Robert Lawlor's *Sacred
Geometry: Philosophy and Practice* both discuss this subject
at length.
------------------------------------------------- From:
Mark Reynolds <marart@pacbell.net>
It can be a risky business to speak of numbers having any
other essence but that essence that quantifies; that is, that
measures and solves problems, mostly scientific. Some professionals
get very upset over the thought that numbers may have other qualities,
almost human-like qualities and possibly even something akin
to personalities with meanings other than purely scientific and
practical. So I would like to refer the reader to a book by C.
G. Jung's close associate, Marie-Louise von Franz. It is called
*Number
and Time: Reflections Leading Toward a Unification of Depth Psychology
and Physics *(1970). It addresses the first four numbers
only:1, 2, 3, and 4, but it deals with them in a way that is
scholarly (many footnotes and references), and also has a scientific
and historical foundation. It is perhaps heady reading for some,
but may well be of value in this vast study of the 'quality of
numbers'.
It should also be remembered that it is not so much what we think
of numbers in the present age, but what numbers were thought
of and how and why they were used at the time of Alberti, the
time Robert is speaking of. Let us not take this out of context
and miss the clarity, perception, and wisdom of Dr. Tavernor's
points.
------------------------------------------------- From:
Mike Bispham <MikeBispham@aol.com>
I take Robert Tavernor's words in two parts.
The first involves practical experience. No matter how much
explanation you offer, there is no substitute for direct hands-on
experience - whether in the solving of geometric problems with
compass and straightedge, or, as Tavener's example - actually
surveying buildings. Its a variant on the theme that builders
and carpenters often offer to architects: "If _you_ tried
to make the damn thing, you'd understand _why_ its a silly idea"!
There's no substitute for _doing_ if you want to gain an intuitive
understanding of the relationships between dimension and form.
Its the difference between a musician and a scholar of music
who can't play a note.
The second is a reference to Pythagorean number mysticism.
I think what you are asking is: "how do we get this rather
strange idea across to modern minds"?
It is perhaps impossible, without a good deal of serious study,
to grasp the point of Pythagorean number mysticism, never mind
the detail. Modern examples? The differences between a single
person and a couple is one that comes to mind - that might be
extended by the addition of a baby - but it peters out there.
Is there really, however, much point; or to put it another
way, can we find, create, a point? These issue are meaningless
in the modern world, except and unless they can serve as pointers
to good architecture. That universe has disappeared; the practices
that accompanied it can serve only to provide historical understanding
and perhaps inspiration for an approach to architecture that
incorporates higher values than cost/return. Such approaches
might retain the idea of harmony, in its modern rather than traditional,
mathematical, sense; between buildings and the social objectives
they serve. Bringing the mysticism along for the ride is a distraction;
relevant only for those that have faith in continuation of its
spiritual premises - a tiny minority - which means that incorporating
it into the architecture may alienate rather than unify.
------------------------------------------------- From:
Alexey P. Stakhov <anna@nest.vinnica.ua>
I think that Leon Battista Alberty borrowed an idea about
"quality" of numbers from Pythagoras who preached:
"A Number is a law and connection of the world, the force
reigning above the gods and mortals", "A Number is
an essence of all things and it introduces to all unity and harmony"
and "All is a Number". Pythagorean number theory was
qualitative. They gave a special attention to the initial numbers
of the positive integers.
The number of 1. The Pythagorean learned that 1 means a spirit,
from which all visible world happens; it is reason, good, harmony,
happiness; it connects in itself even with odd and male with
female. Geometrically 1 is expressed by a point. The Pythagorean
named 1 with "Monada" and considered it as mother of
all numbers.
The number of 2 is a beginning of an inequality, contradiction,
it is a judgement because in the judgement a true and a lie meet.
The number of 2 is a symbol of material atom and geometrically
is expressed by line.
The number of 3. Taking the number of 1, the material atom
2 becomes 3, or movable particle. It is the least odd prime number.
Geometrically 3 is expressed by triangle. The Pythagorean considered
3 as the first true number because it has a beginning, middle
and the end. The Pythagorean considered the number of 3 as a
symbol of alive world.
The number of 4 (4=3+1) is considered by Pythagorean as a
symbol of all known and unknown.
The number of 10. Many people considered this number as a
new unit. The special Pythagorean delight called the fact, that
the sum of the first numbers of the positive integers is equal
to 10 (1+2+3+4=10). Pythagorean named this number as "tetrad"
and considered it as "a source and root of eternal nature".
The number of 36 was the highest oath for the Pythagorean. They
are captivated by the following mathematical properties of this
number: 36 = 1+2+3+4+5+6+7+8 = (1+3+5+7) + (2+4+6+8). As the
number of 36 is formed as the sum of the first four odd numbers
and the first four even numbers Pythagorean made conclusion that
36 is a symbol of the world.
The Pythagorean number theory arose from separation of natural
numbers on even and odd. The Pythagorean doctrine about numbers
was closely interlaced with the doctrine about geometric figures.
They represented numbers as points grouped in geometric figures
and due to such approach Pythagorean came to discovery of so-called
figured numbers. "Triangular numbers": 1, 3, 6, 10,
15, 21, 28, 36, 45, 55, . are their first example. Besides the
Pythagorean considered "quadratic" numbers: 1, 4, 9,
16, 25, ., "pentagonal"numbers: 1, 5, 12, 35, 51, 70,
. and so on.
Besides the figured numbers the Pythagorean introduced so
called "amicable" numbers. So two numbers were named,
each of which is equal to the sum of divisors of other number.
As an example we can consider the number of 220. So-called "own"
divisors of number 220: 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110
give in the sum number 284. But if now we shall count the sum
of the "own" divisors of number 284 (1, 2, 4, 71, 142),
we shall get the number of 220. Therefore numbers 220 and 284
are "amicable".
However there are numbers equalling to the sum of the divisors.
For example, the number of 6 has three "own" divisors:
1, 2,3. Their sum is equal to: 6 = 1+2+3. The Pythagorean considered
remarkable all numbers having such property, and named them by
perfect numbers. They knew three such numbers 6, 28, 496 because:
28 = 1+2+4+7+14; 496 = 1+2+4+8+16+31+62+124+248.
As is well known a number of irrational numbers is limitless.
However, some of them occupy the special place in the history
of mathematics, moreover in the history of material and spiritual
culture. Their importance consists of the fact that they express
some "qualitative" relations, which have a universal
character and appear in the most unexpected applications. The
first of them is the irrational number of root square of 2, which
equals to the ratio of the diagonal to the side of the square.
The discovery of the incommensurable line segments and the history
of the most dramatic period of the antique mathematics is immediately
connected to this irrational number. Eventually this result brought
into the elaboration of the irrational number theory and into
the creation of modern «continues» mathematics. The
Pi-number and Euler's number of e are the other two important
irrational numbers. The Pi-number, which expresses the ratio
of the circle length to its diameter, entered mathematics in
the ancient period along with the trigonometry, in particular
the spherical trigonometry considered as the applied mathematical
theory intended for calculation of the planet coordinates on
the «celestial spheres» («the cult of sphere»).
The e-number entered mathematics much later than the Pi- number.
Its discovery was immediately connected to the discovery of Natural
Logarithms. The Pi and e-numbers "generate" a variety
of the fundamental functions called the Elementary Functions.
The Pi-number "generates" the trigonometric functions
sin x and cos x , the e-number "generates" the exponential
function ex, the logarithmic function logex and the hyperbolic
functions namely the hyperbolic sine and the hyperbolic cosine.
Due to their unique mathematical properties the elementary functions
generated by the Pi- and e-numbers are the most widespread functions
of calculus. That is why there appeared the saying: «The
Pi- and e-numbers dominate over the calculus». The "Golden
Section" is one more fundamental irrational number. The
latter entered science in the ancient period along with the Pi-number.
Hence, dating back from the ancient Egyptian period in the mathematical
science of nature there came into being two trends of the science
progress based on different ideas as to the Universe harmony,
viz. the trend of the Pi-number, basing on the idea regarding
to the spherical character of planets' orbits, and the trend
of the golden section, basing on the dodecahedron-icosahedronical
idea about the Universe structure. The latter idea emerged from
the analysis of cyclic processes within the Solar system and
underlies the calendar systems and the time and geometric angle
measurement systems, basing on the fundamental number parameters
of the dodecahedron and icosahedron, i.e. on the numbers 12,
30, 60 and 360.
Unfortunately in process of its development the classical
mathematics "threw out in garbage can" all Pythagorean
achievements about "quality" of numbers and considered
them as "some funny thing", which cannot be connected
to general number theory.
I think that this fact was a serious mathematician mistaken
influenced on the modern secondary mathematical education and
especially on mathematical education for architects. And just
now a time came to correct this mistaken.
I tried to correct this mistaken in my Virtual
Museum of Harmony and Golden Section.
However the Museum of Harmony and Golden Section is the first
step in the modern mathematical education. My new book "A
New Kind of Elementary Mathematics and Computer Science based
on the Golden Section" is the second step to revive the
Pythagorean mathematics and to bring near mathematics to Nature
and Art. The book is in stage of writing and will consist of
13 chapters and will have about 600 pages. I will inform the
visitors of my Museum about my new book after New Year.
------------------------------------------------- From:
Marco Frascari <mfrascar@vt.edu>
The multi-presence of numbers in architecture ranges from
being measures and units to be symbols, signs, abstraction, cognitive
and perceptual processes, and emblems. Numbers comes not only
from proto-scientific or scientific experiences but also participates
to metaphysical theories, performance of myths, and conception
of nature or the setting of ethical models. Our difficulty to
understand Alberti's view of numbers is that nowadays the mathematical
problem are set by the request of technicians, whereas in past
they were set by demigods as Newton, Euler or Pythagoras and
in the beginning they were set by the God themselves who did
not separate as the technicians do *res extensa* from *res
cogitans*.
The notion of *concinnitas* is one of the most powerful
concepts elaborated by Alberti in his treatise on the art of
cooking … sorry … building. *Concinnitas* is a
powerful tool that architects have for bringing the sensual power
of the *res estensa* within the *re aedificatoria*.
*Concinnitas* usually has been limited to the realm of *res
cogitans*, in particular by some scholars-they cannot help
it: euphemistically speaking, they probably live in a country
where the local cuisine is not very savory. These researchers
have not yet discovered that Alberti, in transferring the concept
of *concinnitas* to architecture, has carried on with it
the ontological essence of its Latin etymological origin. *Concinnitas*
is a quality embodied in the harmony of taste that results in
a properly cooked dish in which the different components are
carefully calibrated. In his treatise, in defying the power of
this architectural quality, Alberti states that *concinnitas*
is *vim et quasi succum* (energy and roughly a sauce). *Concinnitas*
is the sauce in the *tagliatelle al sugo*. Plain cooked
pasta is in itself a meaningless gluey construct; it always needs
a good sauce (*succum*) to put on the force necessary to
enter the realm of the sensuous where architecture and cuisine
are at their best. A culinary exemplum will explain clearly the
use of numbers in non-separating the *res extensa* from
the *res cogitans*.
Among the old dishes of the Italian Piedmont's cuisine, the
king of deserts is the zabaglione (in Piedmont's dialect: *L
Sanbajon*). Fra' Pasquale de Baylon (1540-1592), of the Third
Order of Franciscans, used to suggest to his penitents (especially
to those complaining of the spouse frigidity) a therapeutic recipe
which, summarized in the concinnitas 1+2+2+1, would have given
vigor and strength to the exhausted spouse. Made a saint in 1680
by Pope Alessandro VIII, Santo Baylon became a legend. In the
piedmontese dialect the saint's name is pronounced San Bajon
(o=u). Sanbajon became *Zabaione* o *Zabaglione* in
Italian. To make it beat 1 yolk and 2 teaspoons full of sugar
until the mixture is palest yellow tending towards white, then
beat in 2 eggshells of Marsala wine and cook in a double boiler
(*bagnemarie*). Continue whisking using a hand mixer; do
not let it reach a boil, but remove it from the fire as soon
as it thickens. When it has cooled to merely warm, you fold in
1 egg white beaten until very firm (The recipe, with few corrections
to indicate Sant Baylon numbers, is drawn from Pellegrino Artusi's
*The Art of Eating Well* (*La Scienza in Cucina e l'Arte
di Mangiar Bene*).
------------------------------------------------- From:
Matthew
Landrus <matthew.landrus@wolfson.oxford.ac.uk>
Regarding any possible practical interest in the Pythagorean
mean and extreme proportion during the 15th century, I would
like to add that I have found no trace of this interest in the
work of Alberti, Leonardo, or Pacioli. I've made numerous calculations
with the expectation that the divine proportion may have been
used, but I've found no direct evidence. Without going in to
detail about this, perhaps I could recommend a published study:
Albert van der Schoot's *De onstelling van Pythagoras* (1997).
He argues (in Dutch) that there is no evidence of the use of
"divine proportion" in the works of Renaissance mathematicians.
In English, he's published two of his chapters as "Kepler's
search for form and proportion," in *Renaissance Studies*,
Vol. 15, no. 1 (2001) pp. 59-78. And a German edition of his
book is due out in 2003. Although my findings of the 2/3rds,
root 1, and root 2 preferences are based on the metal-point lines
in Leonardo's drawings, and sources like the floor tiles of Piero
della Francesca's 'Flagellation', (see Martin
Kemp, *The Science of Art: Optical Themes in Western Art from
Brunelleschi to Seurat*, p. 31), I've not taken a close
look at references to Pythagoras by Renaissance music theorists
such as Franchinus Gaffurius. Van der Schoot only refers to Gaffurius
on page 379 of his book, stating that the musician's reference
to Pythagoras has nothing to do with irrational numbers. There
is a reply to this book in *Tijdschrift voor muziektheorie*,
vol. 6, no. 1 (Feb 2001) p. 61-62, by Jeroen van Gessel: "Reactie
op Albert van der Schoot." But I've not had a chance to
read this. Nonetheless, I think that Van der Schoot's book is
quite thorough and very helpful to anyone looking at the problem
of the "quality of number" in Renaissance sources.
------------------------------------------------- From:
Matt Insall <insall@umr.edu>
I am not a philosopher,in the sense of having read or trained
in that subject. However, it seems to me that all mathematicians
and engineers, among others, develop within them a philosophy
of mathematics, and specifically of arithmetic. It is the ability
to communicate aspects of this philosophy to others that distinguishes
those who are referred to as "philosophers of mathematics"
from those who are not so proclaimed. I find it difficult to
address questions such as these, because I do not know Alberti,
or the tradition from whence he hails. Moreover, having not been
trained specifically in philosophy, I do not (yet?) understand
the classical philosophers' use of the term "quality".
Thus, I cannot speculate as coherently as I woul like on the
meaning of "quality of number" in the quoted passage,
except by directly discussing the usage within this particular
passage, as I see it connecting with certain concepts I have
met in other studies.
In computer science, specifically in object-oriented programming,
one assigns to an object certain ``attributes'', which I consider
to be "qualities", in the sense employed in the passage
below,where Alberti is quoted as saying "numbers are not
just abstract things, they describe qualities too". Thus,
an object can be identified by all of its attributes, or qualities.
However, in computer science, one my change the language or terminology
in which the attributes are expressed, and when this is done,
the specification of an object can appear to be significantly
different than in the original formulation. This, however, does
not somehow "cause" the object in question to not be
the "same" object, for, typically, the changes that
are performed are according to certain rules of transformation
that can be reversed. (This is, in logical terminology, basically
a syntactic transform.) When the transformed list of attributes
is transformed back, the original list is recovered, and the
"same" object still has the "same" attributes.
The only way this can be questioned is in dynamic situations,
in which the original object really is removed from the computer's
memory, and may be replaced by another object, for which some
of the previous attributes fail. There are ways, from classical
logic, to model this explicitly, by taking temporal considerations
into account with an explicit time variable, and then to consider
the temporal revisions of the attribute lists from a static language
to be an additional attribute of an object, but the linguistic
complexity gets quickly out of hand. Thus, one models these ``changes
in quality'' in a less explicit manner, via temporal logic representations.
Then the attributes one provides are not exhaustive, and can
conceivably be satisfied by an object not intended to be specified
by that given list of qualities, but in certain circumstances,
that is deemed to be satisfactory for the purpose at hand.
Now, the qualities of an object such as a building are great
in number, and can also be described from multiple perspectives,
so that a minimal, but complete, set of attributes required for
the specification of a building's construction can be difficult
to obtain. Yet, in providing specific numbers as specifications
for that building's construction, one is providing to the builders
a list of attributes for the building that is adequate to accomplish
the task of actually constructing it so that it satisfies a more
abstract set of attributes, such as local building codes, safety
standards, etc. The numbers are therefore not merely abstract
objects in a mathematician's universe, and they are not merely
a linguistic tool, but in the appropriate context, they take
on the role of attribute-specifiers that provide for certain
buildings to have certain qualities that are of importance to
the people who will use them.
The fact that Alberti said "numbers are not just abstract
things" signifies to me that he realizes that there is an
abstract quality to numbers. In engineering, one of the abstract
qualities of numbers is signified by the use and usefulness of
multiple mensuration systems, because the same physical quantity
can be represented by any number, by merely changing the system
of measurement, or the location of a frame of reference. In such
situations, it is the relationships between the numbers that
is preserved by the change from one system of measurement to
another, or from one coordinate system to another, and those
relationships continue to convey to the learned observer the
qualities, or attributes, of the object in question, and so,
even in these abstract interpretations of the numbers involved,
the concrete qualities exist.
------------------------------------------------- From
Bata Tamas <yiu68807@nifty.com>
It is a very interesting question, though I guess the answer
will never be complete. Of course, the majority of qualities
is subjective. I suppose you are looking for those few which
are objective. Apart from architectural qualities that are determined
by culture by culture, there are only a few objective place and
time independent qualities and most of them are platitude, such
as health, fertility, strength..etc. Most of them are related
to the physical well being of human. However, it might be helpful
to ask some physician and biologist that how numbers represent
quality for them. If there is any normality of numbers concerning
health as an example, that normality should be valid for architecture
too.
------------------------------------------------- From:
Richard Mankiewicz
<richard.mankiewicz@ntlworld.com>
The sentiment is undoubtedly neoplatonist, just look at Pico
della Mirandola, a contemporary of Alberti. Also found in Vitruvius.
Briefly, the harmonic relationships on a monochord are given
various human qualities, when those relationships of length are
transferred to proportions in design and architecture, then those
same human qualities are also preserved. The whole branch of
arithmology, numerology, cabbala is then brought in, thus breathing
life into the numbers, and making architecture a kind of solid
music, frozen in space. I think you have numerous examples in
Florence. If the universe is indeed a harmonic place, then the
aim is to mirror that macrocosmic harmony within one's own microcosmic
self. Architecture, and mathematics, are guides towards this.
see Plato, Iamblichus, Boethius etc and nearer to our time, see
the moral geometry of freemasonry.
------------------------------------------------- From:
Gert Sperling
<Gert.Sperling@t-online.de>
I am not able to give reasons for the quality of numbers "by
experience" as Alberti did, but it is possible that Alberti
was confermed in his praxis by the true quality of numbers described
by the ancient Nicomachos of Gerasa in his philosophy and quoted
till to the 18th century in our culture. Nicomachos gave the
numbers quality-connotations and linked them with different ancient
gods, phenomena of nature and human being and forms and types
of geometry, for instance the triangular numbers, the square
numbers, the similar numbers, the perfect numbers and so on.
They also were combined with different sciences like astronomy
and music and most important to create harmony by fusing even
and odd numbers with special qualities.
The source is:
Nicomachus of Gerasa, *Introduction to Arithmetic*, trans.
Martin Luther D'Ooge, with studies in Greek arithmetic by Frank
Egelston Robbins and Louis Charles Karpinski (New York, The MacMillan
Company, 1926. The queried matters you find in the chapters VII-IX,
pp.88 - 128.
------------------------------------------------- From:
Carlos Calvimontes Rojas
<urbtecto@hotmail.com>
In that ancient knowledge can be distinguished two classes
of number: the Idea-Number or Pure-Number, and the scientific
number, with the first being the paradigm of the second, which
is habitually considered a number and which is actually only
a representation, a figure which forgets true numbers. It must
be stated that in general mathematics utilizes models which simplify
the real through regular, conservative conventions (M.C. Ghyka).
Thus it is necessary to think of the authentic numbers, in the
proportions which nature displays, in the symmetries of the stable
and in those which are found in a process of change, in the harmony
of the universe.
------------------------------------------------- From:
Vesna Petresin
<laurent@kosmatih.fsnet.co.uk>
"Numbers are the simplest words", writes Paul Valery.
They order and quantify, yet they are also powerful symbols,
concepts, i.e. qualities; Plato defines number as the essence
of cosmic and internal (personal) stability and the highest knowledge.
Why?
Alberti (just like Aristotle) reflects on sensorial experience
as being an impulse to the thought (De Re Aedificatoria IX, V:
823-35). The meaning is a result of visual and emotional perception
of the observer, not an abstract definition of number and measure.
Numbers carry meaning, they stand for principles, and their two-dimensional
representations are geometric shapes. But numbers can give pleasure
equally to the ear, the eye and the mind (IX, V, 815), so they
must be the basis of both visual and musical harmony, Alberti
argues. Architectural beauty, presumably a highly personal matter,
lies in its geometrical structure, its proportions (i.e. numbers
in relations); defining its criteria makes the concept of beauty
more tangible: numerus, finitio and collocatio sum up to concinnitas.
Alberti's contemporary Cusanus believes numbers are the original
reflection of matter in Creator's spirit, therefore they are
the best means of revealing the Divine truth (Cusanus, Idiota
de mente 6, www III / 524, h V 69, 12). They underlie all beauty
and harmony, and Cusanus argues in De ignorantia (I,5 w I/208,
hI 12, 4 sqq): "If we eliminate number, the differentiation,
order, proportion, harmony and the versatility of the existing
will cease… Sine numerus pluralitas entium esse nequit."
However, the earliest essays on numbers are supposed to originate
from Ancient China, representing number as the key to micro-
and macrocosmic harmony. The notion of cosmic rhythms related
to the number theory can also be found with the Pythagoreans.
Pythagoras believes all things are ordered by numbers; the monad
is the principle of things just like singularity precedes multiplicity.
Monad as an uncountable unit/entity coincides with the notion
of divine infinity. The universe is structured according to numeric
harmony, therefore ideal numeric proportions reflect the unity
of the macrocosm (the Universe) and the microcosm (human spirit).
This could explain why numbers as symbols of universal order
in space and time, creating harmony as well as a relation between
the divine and the human within the universe have been used in
sacred and monumental architecture.
The notion of number as a quality can also be found in Gestalt-based
as well as more recent visual theories, where form is considered
a visual constant, its semantic value being modified by visual
variables such as number, size, weight, location, texture etc.
It is interesting to observe that Carl Gustav Jung understood
numbers as spontaneous, autonomous phenomena of the subconscious
and described them as 'archetypal symbols'.
------------------------------------------------- From:
Nayer Tahoori <n_tahoori@hotmail.com>
Although our lives have been impressed deeply by modernity
but we have preserved our religious beliefs in thinking, so numbers
have never been just as quantities for us. They have symbolic
meanings in our traditional [Iranian] culture from Zoroastrian
to Islam and are sacred for us, as you have Trinity in Christianity.
More over, I am sure that Pythagoras insight about numbers is
well known for you. The source of the most of these is from the
ancient astronomic science and chemistry, Hermetic and mystic
intuitions and finally the order of nature.
Copyright ©2002 Kim Williams
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