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STRUCTURALISM
A. Mackay
To cite this version:
A. Mackay. STRUCTURALISM. Journal de Physique Colloques, 1990, 51 (C7), pp.C7-249-C7-255.
�10.1051/jphyscol:1990725�. �jpa-00231124�
STRUCTURALISM
A. MACKAY
Department of Crystallography, Birkbeck College, University of London, Malet Street, GB-London WClE 7HX, Great-Britain
A b s t r a c t
Structuralism is a significant intellectual movement. Material structure, which underlies all intellectual structure, begins with the geometry of space and the division of space into regions. Recently, developments have taken place in the mathematical world which, with the aid of computer graphics, have enabled the ideas of classical crystallography to be much extended by the incorporation of curved 2-dimensional manifolds or interfaces.
"The human mind has first t o construct forms, independently, before we can find them in things." Albert Einstein.
"There exist only atoms and empty space. All else is opinion". Democritos, (4 cent. BC).
1 Introduction
Stnxcturalism is really an extension to the social and human sciences, such as anthropol- ogy and linguistics, of an outlook which was and is, unquestioned in the natural sciences.
Components plus relationships go to produce functions and evolution. I must confess that I belong to the empirical model-making British school of thought, and do not take very seriously the yarious fashions such as impressionism, idealism, existentialism, realism, surrealism, constructivism, deconstructivism, cubism, or even icosahedrism, which period- ically sweep over the French intellectual scene. However, structuralism seems to me to be one of the more important viewpoints. The main protagonist, Jean Piaget (1968) [g] says:
En premikre approximation, une structure est un systkme de transformations, qui comporte des lois en tant que systkme (par opposition aux proprietks des Blkments) et qui se conserve ou s'enrichit par le jeu mGme de ses transformations, sans que celles-ci aboutissent en dehors de ses frontikres ou fasse appel B des 414ments extkrieurs. En un mot, une structure comprend ainsi les trois charactkres dc totalitk, dc transformations et d'autorkglage.
'To a first approximation a structure is a system of transformat,ions, which comprise laws relating to the system (in contradis- tinction to the properties of the elements individually) and the structure is conserved or enriched by the transformations without travelling outside its boundaries or calling upon external elements. In a word, a structure has the three characteristics of totality, trarisforrnations and self-regulation.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1990725
C7-250 COLLOQUE DE PHYSIQUE
This is rather like a group, such as one of the crystallographic space groups, or a cellular automaton, or a mathematical object like the Mandelbrot set, where the pattern of relationships between the objects is of greater significance than the nature of the motifs themselves.
It was a surprise to the French intellect (and not only to the French 2, that anything so supremely elegant and abstract as Maxwell's equations could come from the empirical model-making British school of science.
"In the beginning, God said:
D E = p/eO V X E = -dB/dt
V . B = O
V X B = j/(eOc2)
+
(l/c2)dE/dt...
and there was light." [4].Also in the beginning, as we read in Genesis, God made an interface and recognised the topological law, which Poincard formulated later, that "The boundary of a boundary is zero".
Pierre Duhem (1861- 1916) complained:
This whole theory of electrostatics constitutes a group of abstract ideas and general propositions, formulated in the clear and precise language of geometry and algebra, and connected with one another by the rules of strict logic
.
Thiswhole fully satisfies the reason of a French physicist and his taste for clarity, simplicity and order
...
Here is a book [by Oliver Lodge] intended to expound the modern theories of electricity and to expound a new theory. In it are nothing but' strings which move around pulleys, which roll around drums, which go through pearl beads...
toothed wheels which are geared to one another and engage hooks.We thought we were entering the tranquil and neatly ordered abode of reason, but we find ourselves in a factory [2].
Systems are made hierarchically out of components and subcomponents; the ways in which the components interact determine how the system operates and evolves.
This view was implicit in the great poem "De Rerum Natura", [On The Nature Of Things - which W.H.Bragg copied as the title of his own book], written by Lucretius almost two thousand years ago, in which he set out his system, seeking (as did the Encyclop~dists
2J.D.Bernal, who was Irish, wrote: "In England, more than in any other country, science is felt rather than thought ... A defect of the English is their almost complete lack of systematic thinking. Science to them consists of a number of successful raids into the unknown". [cf. T.S.Eliot!] ("The Social Function of Science", 1938. p.197).
all properties of matter, including life and consciousness, without supernatural sanctions.
Lucretius propagated the atomism of Democritos whose works had been wilfully suppressed by philosophers such as Plato.
I must immediately anticipate, as Lucretius did, the charge that we are reducing ev- erything to atoms. Reductionism is, I maintain, simply a term of abuse, based on a misunderstanding of science. When we start to construct the wave-function for two parti- cles we find immediately that there is a cross term explaining exactly how a system is more than the linear sum of its parts. The latest development is the demonstration by Alain Aspect that even parts several metres away from each other are connected. The universe is indeed a single system. However, fortunately, many nearly separable sub-systems can be distinguished, and this makes analysis possible. We may consider only the more significant eigenvectors in the whole system.
I must also refute the charge of determinism. Probably the present preoccupation with chaotic systems models our current political climate, just as the laws of conservation and later of evolution paralleled the seventeenth and nineteenth centuries. We realise that even in strictly Newtonian systems, such as a simple damped pendulum driven by a periodic force, may show regimes of chaotic behaviour. Determinate laws do not imply that we can predict the state of the system at a given future time. Laplace's grandiose proposal was quite illusory, as he must himself have well known. Even our Solar system, the model of predictability, is weakly chaotic [6].
Lucretius is vindicated in almost every number of "Nature" [the journal], most signally recently, where single atoms have been manipulated one at a time to spell out, not the names of Jehovah or of Allah, but the initials of IBM [3].
2 Crystallography
The atoms of Democritos were first imaged by W.H. and W.L. Bragg [l] when W.H.Bragg added_pho$~graphica11y the experimentally. determined amplitudes of a_ F~uricr~series and displayed the arrangement of the atoms in crystals of the mineral diopside [CaMySi206].
Since then the structures of tens of thousands of crystals have been determined.
Already in 1931, J. B. S. Haldane in Gowland Hopkins' biochemical laboratory envisaged that life processes were to be understood in terms of the way in which the atoms were arranged in space:
"Write down the structural formula of human type C oxyhremoglobin, and briefly summarize the evidence on which it is based. (Structural formulae should be written stereoscopically. A stereoscope is provided)." Examination question en- visaged by J.B.S.Haldane, (1931).[5].
The great success of the method of X-ray crystal structure analysis, which uses crystals to perform what can be seen, in the light of the Abbe theory of the microscope, to be a kind
C7-252 COLLOQUE DE PHYSIQUE
of microscopy, has diminished the attention paid to other structures which are ordered, but do not follow the exact paradigm of crystallisation.
In the period between the wars, proper attention was paid to other forms of ordering but, with the appearance of the computer, which performed as the computational equivalent of a lens, crystalline substances became greatly favoured objects for study. However, that paradigm is now changing and we are again looking at other forms of ordering, which may not have such si~nple Fourier transforms.
It is now timely to turn our attention more systematically to structures which are not orthodox crystals and also to take the new methods and approaches and apply them back to regular crystals.
Considering bhe many distinguished people present a t this workshop we have a chance to formulate a manifesto for the "new crystallography".
Classical crystallography stands on the geometrical foundations of symmetry the 14 Bravais lattices, the 32 crystallographic point groups and the 230 space groups. We couple with this the geometry of diffraction and the reciprocal lattice.
Further generalisations are needed when we come to consider either random structures or the partially ordered structures suggested by the great variety of structures of lytropic and thermotropic liquid crystals. These lead directly to considerations of the structures encountered in living systeals.
Even Joseph Needham had written:
I regarded the nature of biological organisation as a purely philosophical question, and excluded it from scientific biology.
...
I had not seen the full significance of the analogous science of crystallography. I am glad to have an opportunity of cancelling what I then said.("Order and Life", Cambridge U.P., 1936. p. 17)
In the same book he emphasised the significance of Bernal's insights as to the importance of liquid crystad striict,ures in biology.
Earlier, in 1835, Lobachevskii himself [7] had thought that his "pangeometry" [the non- Euclidean geometry of hyperbolic space]
"...
might find application in the intimate sphere of molecular attraction". So, to some extent has it proved. The metric required by local interactions may not be commensurate with the Euclidean metric of actual space and frustrations may occur.New geometries, involving curved space with non-Euclidean metrics have been applied for the understanding of random and liquid crystalline structures. It is the purpose of this workshop t,o bring t,oget,her people working wit,h such geometries.
3 Geometries
Classical crystallogra~hy is confined to the Euclidean space
E ~ ,
but other worlds can be imagined. In particular, geometry on the surface of a sphere S2, widely used sincesurface of a 4-D sphere [8],[10], H2 and H 3 , hyperbolic spaces and T 2 , toroidal space.
In two dimensional manifolds it is easy to see the problem. For a plane the average coordination number of points in a network must be six. On the surface of a sphere it may be less thatn six and in the hyperbolic plane it may be more than six.
It is now of the greatest interest to try to apply some of the mathematical results of topology [11] t o the problems of spatial structure.
Atoms form a very clear and nearly separable level of structure. A major problem is to pick out nearly separable higher levels. Above molecules it is harder t o find useful constructs.
In many cases, such as synthetic and biological fibres and membranes, units are fairly clear. This workshop will concentrate on membranes and interfaces where surfaces - two- dimensional manifolds - are often distinguishable.
Of the four categories
,
points, lines, surface and volumes, surfaces are the hardest to visualise, particularly if they are overlapping and intersecting, like for example, the structure of the heart. In contrast, we seem to have been provided through evolution with good built-in routines for processing trees - perhaps those of our ancestor who were not so good missed their jumps and did not reproduce.4 Surfaces
The mathematical literature is now very large and, under the influence of physical demands and the rise of computer graphics, interest in surfaces has recently grown rapidly. People here will be describing their own work. However we may separate out a number of aspects.
Minimal surfaces are a major category of mathematical surfaces. They are 'soap film' surfaces and have minimum areas (or a t least stationary areas) and zero mean curvature.
Soap bubbles are an attractive study and their physics and mathematics was popularly demonstrated by, for example, C . V . B O ~ S ~ . The catenoid, the catenary of revolution, sus- pended between two rings, is a fundamental component, and can be found in variants in many of the periodic surfaces. It may be seen biologically as a neck between two mem- branes and indeed it has been proposed that such processes might also occur in kaolinite sheets.
5
Computer Graphics and Virtual Reality
The atomic force microscope has brought us to the point of realising H.G.Wells' story
"The Ring", where a man shrinks and falls between the atoms of a gold ring. Electron
31t was the postulation of a third dimension (distance) which enabled the puzzling movements (such as retrogression) of the planets in the two dimensions of the celestial sphere, to be understood as being the projections of the elliptical orbits in three-dimensional space
4Boys' book was published in 1890 by the Society for the Propagation of Christian Knowledge, which also published books on Crystallography and notably "A Catechism of the Steam Engine" along with evidences for the existence of God.
C7-254 COLLOQUE DE PHYSIQUE
microscopy of various kinds has superceded X-ray single crystal structure analysis as the paradigmatic method of investigation. I t is no longer necessary to pay with the assumption that all unit-cells in a structure are the same for information about any one of them. If there are differences from cell to cell we can see them. In quasi-crystals all unit cells are different.
There are many other instruments for 3-dimensional observation at a range of scales (NMR tomography, etc.). Almost any signal can be used to make pictures which can be observed at one scale and presented at another, not necessarily linearly. There is enormous scope for imagining new methods of presentation.
Computer graphics are one of the principal tools for exploring the possible structures which may be encountered or may be constructible. One of our principle objective must be to improve the means of representing possible worlds and in interfacing them with our minds. Examining possible worlds we can choose which we wish to realise.
All these methods of observation and visualisation depend on geometry. Piaget saw mathematics as the fundamental level of structuration. It is a matter of experiment to determine what part of the mathematics of geometry is actually that of nature.
I think that structuraZism as a philosophy has now a more active aspect. We not only observe the structure of the universe but may participate in trying to re-structure it according to our plans embodying choices among our visions of what is possible
.
During the Thatcher regime in Britain planning has been abandoned in favour of the blind forces of the market5, but perhaps now we may see a return to coherent planning before our time runs out. As in a laser if N atoms act in phase their amplitudes add so that the intensity is hr2. If they act at random the intensity is only N.In the limited field of this workshop we have a chance to re-fashion outlooks on the Geometry of Interfaces into a new coherent field of science.
F'rench culture is characterised by logic, clarity and order. When we approached a certain French town we saw signs marking three roads: The centre road is marked, not unexpectedly, "Centre Ville", The left road is marked "Toutes Directions"; and the right road is marked: "Toutes Autres Directions". To see the clear logic in this we must under- stand that a "Direction" is a destination which has been nominated as a recognised goal.
"Autres Directions" are other destinations not yet so nominated. I think that the task of this workshop is to establish the topic of the geometry of interfaces as a clearly recognised direction of science deserving moral, intellectual and financial attention.
References
[l] W.H.Bragg, Zeit. f. Krist., 70, 488, (1929).
[2] P. Duhem, La Thebrie Physique, Ch. 4,5. (qu. M.B.Hesse, "Models and Analogies in Science", Sheed and Ward, London, 1963. p.2).
5Even Mrs. That.cher sometimes feels the lack of a theoretical basis and asked, at the Conservative Philosophy Group, "The opposition have an ideology against which they compare their policies. We must have one too".
[4] grafito - seen in Cambridge ca. 1970.
[5] J. B. S. Haldane. [Question in a mock exam paper looking forward 25 years from 19311. Brighter Biochemistry, Cambridge, 1931. Reprinted TIBS, March 1981, p.91.
[6] J. Laskar, "A numerical experiment on the chaotic behaviour of the Solar System", Nature, 338, 237-238, (1989).
[7] N. I. Lobachevskii, Noviye nachala geometrii s pollnoi teoriei parallelnykh, (1835-38).
[8] Jour. Phys., (A), 13, 3373-3379, (1980).
[g] J. Piage t
,
Le Struc turalisrne, Presses Universitaires de France, 1968.[l01 J-F. Sadoc (ed.), Geometry in condensed matter physics, World Scientific, Singapore, 1990. pp. 500.
[l11 W.
