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Query: The Quality of Number

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?

Send an e-mail to respond to this query

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>

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.

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.

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.

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