ORIGINAL QUERY:
Date: Wednesday, 27 November 2002 11:27:42 +0100
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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?"
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Comments
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.
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.
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:
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.
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.
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.
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. :-)
Also, see Keith Critchlow's book, Islamic Patterns: An Analytical and Cosmological Approach.