Read here about a philosophical issue in mathematics: how arithmetization and tensegrity studies conflict.
Arithmetization is the conception of the real world in terms of numbers. While the overall trend is as old as history itself, the systematic replacement of geometric and topological reasoning with numerical methods is actually quite new.
The topic is important to tensegrity researchers, since tensegrity is predominantly a geometrical way of thinking. The discovery of tensegrity constructions in the 20th century is part of a small but significant trend back to geometrical ways of thinking.
Selected Quotes Against Arithmetization By Tensegrity Researchers
Motro wrote that his discovery of tensegrity in the work of Emmerich "was for me the equivalent of a life-line after so many years of equations and rationality". To read more, see selections of his book online.
Fuller wrote, "I determined then and there to seek out the comprehensive coordinate system employed by nature. The omnirational associating and disassociating of chemistry__always joining in whole low-order numbers, as for instance H '2' O and never H 'pi' O persuaded me that if I could discover nature's comprehensive coordination, it would prove to be omnirational despite academic geometry's fortuitous development and employment of transcendental irrational numbers and other "pure," nonexperimentally demonstrable, incommensurable integer relationships." (Synergetics 410.02). To read more, see Synergetics online.
Annotated History of Arithmetization
The term arithemtization can be traced back to the arithmetization of analysis, a research program carried out in the second half of the 19th century. Kronecker originally introduced the term //arithmetization of analysis//, by which he meant its constructivization in the context of the natural numbers. The meaning of the term later shifted to signify the set-theoretic construction of the real line. Its main proponent was Weierstrass, who argued the geometric foundations of calculus were not solid enough for rigorous work.
//Weierstrass advocated strongly against the building of physical models, and the publication of geometrical insights. In his view, only arithmeticalized insights were valid. This is one reason that many tensegrity and geometrical innovations were propagated by non-mathematicians such as Fuller, Snelson and Emmerich. Even today, mathematical publications on tensegrity are mostly calculus-based arithmeticalizations of tensegrity models//
The highlights of the arithmeticalization research program are:
- the various (but equivalent) constructions of the real numbers by Dedekind, Cauchy and Weierstrass resulting in the modern axiomatic definition of the real number field;
- the epsilon-delta definition of limit; and
- the naïve set-theoretic definition of function.
//Numbers as written today, such as 932.1, are actually polynomials. Each number is an expression of finite length constructed from constants (or variables) using only the operations of addition, and multiplication by non-negative, whole-number exponents. The number 932.1, for example, is a tidy notation for the more unwieldy (9x10^3) + (3x10^2) + (2x10) + (2x10^-1). So what, you may ask. Well, it is impossible to express most numbers in this form. It is only by calculating limits of functions, using methods of the calculus, that real numbers can express precisely the most elementary and simple geometrical facts.//
An important spinoff of the arithmetization of analysis is set theory. Naive set theory was created by Cantor and others after arithmetization was completed as a way to study the singularities of functions appearing in calculus.//
The arithmetization of analysis had several important consequences:
- the widely held belief in the banishment of infinitesimals from mathematics until the creation of non-standard analysis by Abraham Robinson in the 1960s, whereas in reality the work on non-Archimedean systems continued unabated, as documented by P. Ehrlich;
- the shift of the emphasis from geometric to algebraic reasoning: this has had important consequences in the way mathematics is taught today;
- it made possible the development of modern measure theory by Lebesgue and the rudiments of functional analysis by Hilbert;
- it motivated the currently prevalent philosophical position that all of mathematics should be derivable from logic and set theory, ultimately leading to Hilbert's program, Gödel's theorems and non-standard analysis.
//Note above, "the shift of the emphasis from geometric to algebraic reasoning"; tensegrity structures are by their nature geometric forms, and should be approached first by geometric reasoning.//
The Basis of Arithmetization
Lakoff and Nunez in "Where Mathematics Comes From" detail at great length how arithmetization can be analyzed as a chain of metaphors springing from geometric and biological reality. In their "Case Study I, Analytic Geometry and Trigonometry", they list many metaphorical steps that may be summarized as follows:
- Functions are treated as numbers.
- A circle is regarded as a set of cartesian coordinate pairs
- The unit circle, with radius as unit 1, is taken as the basic or archetypal circle
- The ratios of triangles, as codified in sin, cos, tan, are regarded as lengths of various constructions within the unit circle
The result is, in tensegrity structures, physical angles are confused with the polynomial value of angles as measured, though the value cannot be expressed in any polynomial. See for example in the chart below, how the most elementary geometrical relationships require surds that are allocated symbols rather than polynomial expressions.
Fuller's Resistance Of Arithmetization
Fuller's geometric system, which helped discover tensegrity, is continuous with all previous geometric investigations. But Fuller firmly rejected arithmetical methods and always began his explorations geometrically.
Rejection of Cartesian Coordinates
For example, Fuller rejects that space is best conceptualized as sets of Cartesian coordinates. He accepts that Descartes' mapping of orthogonal axes, as expressed in number tuples (pairs or triplets) does provide a useful mapping of distinct sightable objects in space. Fuller was thrilled with Nasa's space programs and never suggested that they drop their linear equations. Fuller did urge NASA and anyone that would listen to reject the notion that space is ordered in number tuples. In fact, Fuller suggests that thinking this way misleads researchers.
Rejection of Number Set Continuity
Fuller rejects that the continuity of space is best conceptualized as the continuity of the number line. Fuller understands the number line, though he shares an abiding suspicion of irrational and transcendentals with Plato. But numbers are, well, just that: numbers. They are not reality, they emerge from reality, from human brains. Fuller holds that the continuity of space is based on an alternation of co-dependent co-originating co-arising pairs such as concave/convex, event/novent, space/matter and tension/compression. Numbers cannot fully express this alternation, since they irrevocably uniform--they are all NUMBERS. This is one reason that tetra/octa alternation so deeply excited Fuller--it seemed a better model for the alternating nature of reality.
Rejection of Number's Lack Of Center
Numbers are poor expressions of the "centeredness" of reality. Reality, and particular living reality, presents itself in dynamic vivid centers surrounded by non-centers. To comprehend this assertion, consider for a moment your own living body, or the urban plan of a vital, living public space. (See Alexander for further discussion). Arithmetic conveys the opposite message--in it, any number is the same as any other.
In his search to find centeredness in traditional Hindu-Arabic polynomial notation, Fuller returned to the integers repeatedly, and found some satisfaction in the 1-2-3-4-3-2-1 pattern that is found in tossing out nines. He called it, "Indig octave system and zero-nine wave pattern"--see Indigs.
Fuller also struggled to identify centered models of reality. This helps explain his profound excitement when exploring nucleated closest sphere packed models, what he called vectore equilibria.
Rejection of the Dimensionless Point Metaphor
The idea of a point as a dimensionless object was anathema to Fuller's way of thinking. Fuller often spoke about how itwas an inadequate model of the operational, actual behavior of points in reality. Note that the contemporary idea of a dimensionless point is NOT ancient. The discrete point of Euclid's point/line/plane as we know it today was really elaborated in the arithmetized calculus of Weierstrass, just a century ago. Assigning it a dimension of zero, to form a convenient integer continuity with a line (dimension 1), plane (dimension 2) and space (dimension 3), is in reality the construction of an elaborate metaphorical model where exponents are mappings from a number line to a multiplicative result. While profound and beautiful, Fuller constantly warned that this metaphor was also stifling. He argued that the idea of dimension is too useful to be relegated to a synonym for a exponential mapping, and claimed that the critical dimensions of reality were the four sides of the tetrahedron or the six directions of its edges.
Rejection of the Abstraction Necessary in Limit Theory
Fuller was against replacing physical models with theoretical limits that were not fully comprehendable by human senses. Our number system is so inherently crippled that even "one third," a simple ratio, cannot be expressed in decimal notation without an infinite limit (0.333333_). The arithmetized thinker relies too heavily on infinity, Fuller contends, for the most simple conceptions, and so inculcates a debilitating habit of mind that leads the typically arithmetized university graduate to overlook many highly nurturing, compatible and fruitful design solutions to contemporary problems.
In this antagonism to limits Fuller was not alone. James Pierpont of the American Mathematical Society, was also uneasy with the arithmetization agenda promoted by Weierstrass and others. Pierpont spoke of these extreme arithmetizations at Yale University in February 1899:
"The mathematician of to-day, trained in the school of Weierstrass, is fond of speaking of his science as "i c die absolut klare Wissenschaft. ' ' Any attempts to drag in metaphysical speculations are resented with indignant energy. With almost painful emotions he looks back at the sorry mixture of metaphysics and mathematics which was so common in the last century and at the beginning of this. The analysis of to-day is indeed a transparent science. Built up on the simple notion of number, its truths are the most solidly established in the whole range of human knowledge. It is, however, not to be overlooked that the price paid for this clearness is appalling, it is total separation from the world of our senses."