In stark contrast to the Hilbertian project, the Frenchman Henri Poincare dedicated himself to what he called ‘problems that are formulated’ as opposed to ‘problems that we formulate’. Ribbing along the other way enhances the perception that the bottle turns in on itself. Philosophy seized on this and declared that reality was an imperfect image of this exact world: the tree trunk became a defective cylinder, and the plank an adulterated plane. Hence, statements about mathematical objects such as Euler’s theorem (the number of vertices and edges minus the number of faces of the polyhedron is equal to 2), seem to present themselves to mathematicians who merely ‘discover’ their properties. Throughout its long history, the bond between mathematics and the sensible world has always fluctuated between three key philosophical attitudes: representation, construction and simulation. It is also unbelievably diverse. In 1931, Kurt Godel showed that any coherent axiomatic system, a system without contradictions, features propositions that cannot be proven true or false; while other mathematical results from set theory reinforce the sentiment of the ‘unreality’ of the world of mathematics. Yet what characterises a fiction is not so much its logical status (whether it is true or false), but its cognitive function. The 17th century witnessed the concurrent birth of infinitesimal calculus and the science of movement, bringing radically different conceptions of mathematics to a head. Figure 6. In 1872, studying the properties of figures left invariant by a given class of transformations, Felix Klein reorganised the body of all known geometries. Hilbert was the first to fully consider space as a mathematical concept, rather than as the site of our experience, declaring geometry a formal science rather than a set of propositions about ‘reality’, and breaking, for the first time, the bond between this foundational discipline and the sensible world. However, while the universality of mathematical objects does speak in favour of some kind of realism, the discoveries of logic in the 20th century have made this position difficult to defend. The historian not directly immersed in creative mathematical activity (nor prone to making philosophical statements) will contend that the historical and cultural contexts are key to the emergence of certain concepts. The Mongean representation of the drawing plane demonstrates a technique unchanged since the late 18th century

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n, TI. Practitioners of mathematics frequently express a strong subjective sense of discovery or of the exploration of a terra incognita that is already extant and consistent. Harold Tan, Stereotomic Self-Portrait, Diploma Unit 5, Architectural Association, London, 2003

In this particular narrative, the deployment of stereotomic projection to subtract portions of matter from a solid cube offers a metaphor of the application of colour inversion within the cubic red-green-blue (RGB) space of the additive light colour model (a slice of which we casually experience when using the ‘colour-picker’ of digital imaging software). Several philosophical responses are conceivable, which have more or less always coexisted, and are the focus of this introductory essay to the subject. Cara Liberatore, Where the Truly Formless Lives, Superficial Spaces, Harvard Graduate School of Design, Cambridge, Massachusetts, 2010

Analytic geometry at work: a coarse quadruple parametric ripple with sectional parametric curves. The Paris-based scientific historian Amy Dahan-Dalmedico asks why a knowledge of numbers, algebra and abstract forms should be key to our understanding of the sensible world. The philosopher Hartry Field went as far as proposing that mathematical theories are literally false since their objects do not really exist. To produce theories we must envision hypotheses and explore their consequences; we must build models of phenomena, and formulate idealisations. The Wenninger model is of great help for assembling all parts in the right order. A.02 where the Truly Formless Lives GSD2404 L. Figure 1. In conclusion, is mathematical realism independent of sensible reality? The development of the neurosciences has hardly made a difference in this regard. Concerned with methodological rigour and the need for generality, Rene Descartes did not see curves as spatial realities for their own sake, but as the set of solutions to a given equation. The weft outlines the notional continuity between interior and exterior for which the Klein bottle is famed. By the middle of that century, numerous constructions of Euclidean models of non-Euclidean geometries provided these discoveries with intuitive support and accelerated their ultimate acceptance. The Foundations owes its radical novelty to its capacity to integrate the general with the technical and mathematical philosophy with practice. Working independently during the first decades of the 19th century, Nikolai Lobachevski and Janos Bolyai reached the conclusion (already intuited by CF Gauss after 1813) that a geometry of space did ‘exist’ and conform on many points to Euclidean geometry, with the exception of the so – called Fifth Postulate (through a point outside of a line pass an infinity of parallels to that line). Because of the hegemony of the form inherited from Greek mathematics, it took some time to recognise the great diversity of forms of mathematical expression. On the other, a will to conquer and the desire to force one’s way down new paths. Stefano Rabolli-Pansera, Stereotomic Self-Portrait, Diploma Unit 5 (Engineering the Immaterial), Architectural Association, London, 2003

Descriptive geometry at work, submitted in response to an undergraduate design brief on self-representation. This warm-up piece was submitted in response to a question on form and formlessness. LEGENDRE LIBERATORE Cara

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The Riemann sphere (or extended complex plane in algebra and analysis) is the sphere obtained from the complex plane by adding a point at infinity. Is mathematics to be found in the world, or in our minds and imagination? Matthew Chan, Stereotomic Self-Portrait, Diploma Unit 5, Architectural Association, London, 2003

Several successive derivations of stereotomic projection (projective geometry applied to stone cutting) result in a highly articulated prismatic object recording, in metaphorical and narrative terms, an argument between the author and his girlfriend. Constructivism posits that all truth is constructed, without reference to whatever it conforms to. From these new methods, other mathematicians have been able to develop current theories of complexity and chaos. For Austrian philosopher Ludwig Wittgenstein, for example, the mathematical understanding of a statement does not exist outside of its proof, and in this sense the mathematician is merely an inventor (as opposed to a discoverer) who establishes connections and forms descriptions, but does not describe real facts.2

Mathematics is certainly characterised by an impressive coherence, seemingly superhuman, that distinguishes it from other human practices. A powerful research movement converged towards the axiomatisation of geometry, led by David Hilbert and his Foundations of Geometry (1899).3 Beginning with undefined objects whose nature hardly mattered – points, lines, planes or chairs, tables or spoons – Hilbert specified relations between them through axioms, or given rules, no longer seeing the elements of geometry as intuitively realistic objects, but as variables of a formal language (or their symbols). As representations whose purpose is less to provide an accurate description of reality than to help us imagine possibly unreal situations, fictions are assisting our imagination, and that

is where their scientific pertinence lies, science being more than a mere collection of facts. The Ancients did use some analogous curves to explore intractable problems (for example, the duplication of the cube), but discounted such findings as unworthy of what they would have hoped to obtain using a ruler and a compass. Figure 2. J П 2 TI 2

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iThreadj,! Figure 8. By identifying curves in space with binary quadratic forms, Klein advocated the merger of geometry and algebra, now an established tendency of contemporary mathematics. The argument is without a doubt impossible to settle and, unless one is a philosopher, a bit of a waste of time: the question of mathematical realism is a metaphysical problem, forever unanswerable or irrefutable. With Galileo, geometry entered the world and became reality itself. Evolving and morphing as a discipline, it has covered a diverse range of practices and theories. According to this realist point of view, mathematical objects really exist and the ‘great book of the universe is written in the language of mathematics’.1 In this sense, applied mathematics is verified by experience and constitutes an inherent and indispensable part of the empirical sciences. From the latter’s point of view, mathematics should not only be a well-ordered ceremonial, but a tool for controlling and creating things. Yet, strikingly, such diverse forms are comparable, and it is possible to rigorously reinterpret the mathematical knowledge of one civilisation within the context of another. Such a deterministic history may be useful to the mathematician and the student, but it does not take into account the highly contingent aspects of the formation of mathematical knowledge, with its generally chaotic course and winding paths of discovery. Furthermore an understanding of mathematical reality inscribed in history, space, time and culture imbues the work needed to reconstruct and analyse mathematical practices with veritable meaning. For the defenders of an ideal (or Platonist) realism, the brain may possess structures that give it access to the independent universe of mathematics. As she reveals, mathematics, like the world itself, has shifted and fluctuated – over time since its earliest origins in ancient Egypt. Mathematics are not a stable and well-defined object, but rather a plurality of objects, practices, theories, cognitive and collective constructions, which have a history and great diversity. They must compromise between a dimension of historicity proper to mathematical knowledge and mathematics’ potential objectivity and universality. The fact that mathematicians often reach the same results with similar objects through separate paths greatly incites them to believe in the independent existence of these objects. Thus, in constant interaction with one another, representing, constructing and simulating form three inseparable moments of mathematical activity: to represent an object, other objects must be constructed from it; to construct an abstract object (a function, a transformation), we must represent it; and to simulate an object, we must elaborate fictitious representations which mobilise more constructions at will. Why does the knowledge of abstract objects, such as numbers or algebraic structures, help us better understand, control and master the sensible world and its forms? In 1822, however, Jean-Victor Poncelet explored the properties of figures remaining invariant by projection with the systematic help of infinite or imaginary elements, endowing projective geometry with a far greater degree of abstraction and generality. It expressed the very nature of mathematics, torn between two opposed yet inseparable trends: on the one hand we have rigour, an economy of means and an aesthetic standard.