From Internationale Situationniste #5 (December 1960).
The thoughts and observations about it are entirely new; the citations have not been made before; the subject is of extreme importance and has been treated with infinite arrangement and clarity. It has cost me a great deal of time, and I pray that you will accept it and consider it as the greatest effort of my genius.
— Jonathan Swift, Irrefutable Essay on the Faculties of the Soul
If, as Lemaître says, time was an extraneous notion to the situation, situology will be as much a study of the unique identical form, as morphology. But it could rightly be said that situology is a morphology of time, since everyone is agreed that topology is defined as the continuity which is the non-division in extension (space) and non-interruption in duration. The morphological side of situology is included in this definition: that which concerns the intrinsic properties of figures without any relation to their environment.
The exclusion of singularities and interruptions, the constancy of intensity and the unique feeling of the propagation of the processes, which defines a situation, also excludes the division in several times, which Lemaître pretends are possible. But the confusion of ideas by an unlettered person like Lemaître is much more pardonable than that which prevails amongst professional topologists; and which obliges us to distance ourselves from the purely topological terrain to invent a more elementary situology. This confusion is introduced precisely in the formula of orientability which, in reality, is only adaptation to the dimension of time. E.M. Patterson explains that
"the idea of orientability derives from the physical idea that a surface could have one or two sides. Let us suppose that around each point of a surface — with the exception of the points at the edge (boundary), if there are any — a little closed curve is drawn in a defined sense, having been attached to this point. At this moment, the surface is called orientable if it is possible to choose the sense of the curves, of the manner to which it would be the same for all the points sufficiently close to each other. If not the surface is called non-orientable. All surfaces with only one side are non-orientable."
This mixture of geometry and physics is quite out of order. It is easy to prove that a sphere only has one surface, and likewise a ring. That a cone possesses two surfaces and a cylinder three, etc. But logically a surface can only have one side.
Anyway, a surface with two sides is not topological, because there is a rupture in continuity. But the reason for which we are put on the false scent of the double surface with two sides is clear: it's because that's what allows the linkage of topology with the general tendency of geometry: the search for equalities, or equivalencies. Two figures are explained as being topologically equivalent, or homeomorphs, if each can be transformed into the other by a continuous deformation. This is to say that there is a single figure in transformation: situology is the transformative morphology of the unique.
The gravest error which was introduced by adapting the classic perspective of geometry to topology, is the adaptation to classic distinctions of topology of surfaces and the topology of volumes. This is impossible and ridiculous if elementaries of situology are understood, because in topology there is a precise equivalence between a point, a line, a surface and a volume whereas in geometry there is an absolute distinction. This confession is clearly reflected in the Moebius strip, which is said to possess "two surfaces without homeomorphy" or to represent "surfaces with a single side" without a back or front, without an inside or outside. This phenomenon can even lead people to imagine that the Moebius strip only possesses a single dimension, which is completely absurd, because a Moebius strip cannot be made with a piece of string, even less with a line. What is most interesting about the Moebius strip is exactly the relationship between the two lines of the parallel edges.
It is possible to study geometric equivalencies, congruencies and likenesses of a Moebius strip, if a particular fact is taken into account: the length of a Moebius could be infinite compared to this width. It's up to the mathematicians to construct and calculate the Moebius strip at its minimal limit. Once constructed, it would be found that we are dealing with an object where the line which marks the width of the strip at a point taken by chance, makes a perfect right angle with same line drawn on the opposite part of the strip, however these two lines are parallel, if the strip is smoothed into a cylinder. The same line which at one point represents the horizontal at another point represents a vertical. There are thus three spatial dimensions, apart from the space if the strip is flattened. Hence the strangeness of the Moebius strip. Two Moebius strips of this type can thus always be put into likeness, and with the same width of strip, put into congruence.
It seems that no-one has yet remarked on the strange behavior of all topological forms and figures in their relationship with the system of spatial co-ordinates (vertical, horizontal, depth) in which they play, making them be born and disappear, and transforming one into the other. For Euclidean geometry, the system of co-ordinates is a given basis. For situology, no, as it creates and disposes of the co-ordinates at will. Thus Euclidean geometry has a duty to go beyond all situological considerations to take as a point of reference the system of co-ordinates at right angles which is the schema of the law of least effort. René Huygues shows, in his work Art and Man, that it is with the development of metallurgy, after the agrarian epoch, that the division is produced between the two styles of Hallstadt and La Tene, which is none other than the division between geometric and situlogic thought. Through the Dorians geometric thought was implanted in Greece, giving birth to rationalist thought. The contrary tendency wound up in Ireland and Scandinavia.
Walter Lietzman notes, in his work Anschauliche Topology:
"In art, for example in the age of the Vikings, knotwork was used as ornamentation with pleasure. I have before me a photo of the knot gardens of Shakespeare at Stratford, in which the arrangement of flowers in the form of knots is shown… What does Shakespeare see in these knots? I'm not able to say. Perhaps it's a matter of some error or more a deliberate confusion with the theme of the labyrinth. The question is raised twice with him: In Midsummer Night's Dream (act II, scene 1), and in The Tempest (act III, scene 3)."
There is no possible mistake. James Joyce in Finnegans Wake, by pronouncing the absurd phrase "No sturm, no drang", had overcome the ancient conflict between classicism and romanticism and opened a ski-slope towards the reconciliation of passion and logic. What is needed today is a thought, a philosophy and an art which conforms to what is projected by topology, but this is only realizable on condition that this branch of modern science is returned to its original course: that of "the situ analysis" or situology. Hans Findeisen, in his Shamanentum, indicated that the origins of shamanism, which still survives amongst the Lapps, are to be found in the cave paintings of the ice age, and it is enough that the ornamentation which characterizes the Lapp presence is simple knotwork. The knowledge of secret topologies has always been indicated by the presence of signs of knots, strings, knotwork, mazes etc. And in a curious way since antiquity the weavers has transmitted a revolutionary teaching in forms which are more or less bizarre, mystifying and subverted. A history too well known to have been studied seriously. The perversion in that should be noticed rather than the reverse.
The relation that the writings of Max Brod established between Kafka and the Danish astronomer Tycho Brahe is as profound as the relationship between Shakespeare and Hamlet: and their presence at Prague which, since the time of La Tene radiated topological thought, is as natural as the astonishing results that Kepler could extract from the calculations of Brahe, by adapting them to the methods of geometry and classical mathematics, which was impossible for Tycho Brahe himself. This shows once more that topology remains the source of geometry, and that the contrary process is impossible. This indicates the impossibility of explaining the philosophy of Kierkegaard as a consequence of the philosophy of Hegel. The influence of Scandinavian thought in a European culture is incoherent and without permanent results, like a true thought of the absurd. That there has always been a Scandinavian philosophical tradition, which structures the tendency of Ole Roemer, H.C. Oersted, Carl von Linné etc., completely distinct from English pragmatism, German idealism and French rationalism is a fact which can only be astonishing in that it has always been kept secret. With the Scandinavians themselves ignoring the base logic of this profound and hidden coherence, it is as much ignored by others. I have the greatest mistrust of all the ideas on the benefits of learning. However in the actual situation in Europe it seems to me that an ignorance of this subject presents a danger. Thus I consider that the fact that Swedenborg and Novalis has been mine engineers is more important than the chance postulates of such as Jaspers which allowed the label of mad schizophrenics to be stuck on their backs. This is not because this is a fact which could be established in a scientific manner, but because it is a basic skill of topological thought, like that of weavers, and this fact could lead us to the precious observations for the founding of situology.
But all this is only presented as a possible technique subordinated to the work of the SI, the allies and enemies of which can easily be seen. The situationists reject with the greatest of hostility the proposal arising in Bergier and Pauwel's book, The Dawn of Magic, which asks for help in setting up a proposed institute to research occult techniques; and the formation of controlling secret society reserved for those today who are in a position to manipulate the various conditions of their contemporaries. We should not in any case collaborate with such a project, and we have no desire to help it financially.
"From all evidence, equality is the basis of geometric measurement" as Gaston Bachelard said in Le Nouvel Esprit Scientifique. And he informs us:
When Poincaré had shown the logical equivalence of various geometries, he stated that the geometry of Euclid would always be the most useful, and that in case of conflict between this geometry and physical experience, it was always preferable to change physical theory than change the elementary geometry. Thus Gauss had pretended to experiment astronomically with a theorem of non-Euclidean geometry: He wondered if a triangle located in the stars, and hence of enormous surface, would show the shrinking of surface indicated by the geometry of Lobatchowski. Poincaré did not recognize the crucial character of such an experience.
The point of departure of situography, or of plastic geometry, must be Situ analysis developed by Poincaré, and pushed in an egalitarian direction under the name topology. But all talk of equalities is openly excluded, if there aren't at least two elements to equalize. Thus the equivalence teaches us nothing about the unique or the polyvalence of the unique, which is in reality the essential domain of situ analysis, or topology. Our goal is to set a plastic and elementary geometry against egalitarian and Euclidean geometry, and with the help of both to go towards a geometry of variables, playful and differential geometry. The first situationist contact with this problem is seen in Galton's apparatus that experimentally produced Gauss's curve (see the figure in the first issue of Internationale Situationniste [in The Situationists and Automation]). And even if my intuitive fashion of dealing with geometry is completely anti-orthodox, I believe that a road has been opened, a bridge thrown across the abyss which separates Poincaré and Gauss as far as the possibility of combining geometry with physics without renouncing the autonomy of the one from the other.
All the axioms are cut offs against the non-desired possibilities, and by this fact contains a voluntary illogical decision. The illogic which interests us at the base of Euclidean geometry is played between the following axes: things which are superimposed upon each other are equal; the sum is greater than the part. This absurdity is seen, for example, the moment we start to apply the definition of a line as length without breadth.
If two lines are superimposed, one equals the other. This must result in either two parallel lines (which shows that the equality is not perfect and absolute, or that the superimposition is neither) or the union of the lines in a single line. But if this line is longer than a single line, or if it has acquired width, the lines would not be equal. But if the lines are absolutely equal, the whole is not bigger than the part. This is an indisputable logic, but if it is true, we are in an absurdity because geometric measurement is precisely based on the axiom that the whole is greater than the part. The idea that two equal lengths are identical is found in geometric measurement. But two things can never be identical, because then we would say they were the same thing. If a murderer must be identified to a judge, it isn't enough that this is an individual who looks exactly like the person who committed the crime. The identical will not do in these circumstances. It is certain that there are no equalities, no repetitions, as in the case of the Konigsberg bridges. In geometry, an identity of length and position excludes all quantitative consideration. But how is it possible through superimposition to reduce the infinite number of lines of equal length to one line, which is no bigger than any single line of these; in such a case where it is unthinkable to divide a line in two, are both equal to the divided line?
If a line is moved from its position, at the same time it remains in its position, a surface has been created rather than two lines. The superimposition, which shows that the two lines are equal, cannot be practiced without the duality disappearing: otherwise they could not be equalized. A single line is equal to nothing. This proves that there is no reality in the absolute idealism of Euclid's formula that a line has no thickness. The proof by superimposition is impossible, even if the process is modernized by employing the formula of congruence, or an identity of form, but still excepting spatial position.
We can reduce a thousand points to a single point by superimposition, and this point is equal to one of the thousand points. But a point cannot be multiplied and left at the same place, and displaced at the same time. This would be a line. As for volume, these can only be superimposed in the imagination. It could only be achieved with two phantom volumes without real volumes. This abstract character is at once the strength and weakness of Euclidean geometry. The slightest abstraction in topology is only a weakness.
A thousand times zero is only zero, and nothing can be abstracted from zero. Euclidean geometry is used in this irreversible and unilateral sense: it's oriented. And all the geometries, apart from situography, are the same as it. Orientation is a linear concept, and a vector is also called a half-vector, because it also signifies the distance covered, and the sense in which this has been chosen, is called its positive sense. The zero point, chosen at some point on the line is fixed as a point of commencement. An oriented straight line is thus not a line in itself, but the combination of a line and a point. An oriented plane is a plane in which is chosen a sense of rotation called direction, and this plane is also linked to a point, the center of rotation, which could allow the establishment of an axis of rotation at right angles to the plane of rotation.
Space is oriented as there is a sense of rotation associated around each axis of space, called the direct sense of space. This installation allows everything that can be called measurement. But of what does measure consist? This is the most curious thing about this business. All the measures of equal units whether of length, of size, height, mass time or whatever unit derived from these basic notions, consists of their indication by on a half-line, spatial demi-dimension divided into equal intervals oriented from a point of origin towards infinity. This half-line does not need to be straight, but could be inscribe on the circumference of a circle. If the extension makes several revolutions these become the distances of a greater linear or circular extension. Here is the principle to which all possible measure arrives in the final analysis. Any measure cannot explain whatever may be outside of this limit of a development along a demi-line.
Euclidean and analytical geometry were developed within its classical discourse, itself following the orientation of a demi-line. Starting with a point without spatial dimension, this is moved forward and so traces a line. The line is moved forward in a direction perpendicular to its extension to produce a surface, with which the same process is used to create a volume. But this oriented movement, which from a point produces a line, a surface, a volume, this movement in itself does not enter into geometric considerations in its relations with spatial dimension. The inconsistency is evident. The act of superimposition is also impossible without movement, but from the moment when all the necessary movements to establish classical geometry are put on trial, purely spatial phenomena can no longer be spoken of, and nevertheless movement is there from the beginning. We can wonder whether time has only a single dimension, or whether in the future we might not be obliged to apply to time at least three dimensions to be able to arrive at more homogenous explanations of what has happened. That remains to be seen. But one thing is certain: time cannot be reduced to a demi-dimension or to an oriented length with a measuring instrument. We thus also reach another question as to whether what we know as 'time' in its scientific definition, as a measure of duration, and the form under which time enters relativity theory, isn't simply the notion of orientation or a demi-line.
Oriented geometry can, thanks to its orientation, ignore the notions of time inherent to its system. But, in order to take consciousness of the role of time and of its real role in relation to the three spatial dimensions, we are obliged to abandon the path of orientation to demi-line, and to found a unitary homeomorphism.
When we want to use the expression dimension, we are immediately faced with the problem of its exact interpretation and definition. A dimension can be defined in a logical fashion as an extension without beginning or end, neither sense nor orientation, an infinity, and it's just the same with the infinity in the dimension of time. This is eternity. The extension of one of the three spatial dimensions represents a surface, an extension without beginning or end. If the system of linear measurement can only measure the demi-line, the system of measurement from two co-ordinates at right angles can only give a measure of space for figures drawn in a quarter of a surface, and the information of 3D measurements are even poorer as they are drawn within an eighth of a sphere from the angle of measure of 90° of three oriented co-ordinates in the same direction. To avoid this perpetual reduction of knowledge, we shall proceed in the inverse sense.
For the witness of a crime, identification is to define the suspect as the possible unique. But homeomorphism poses us various problems. It could easily be viewed as follows: now it is no longer a matter of identifying the assassin, but the poor victim that the brute has voluntarily ridden over several times with their motor car. They have an aspect, which differs in a tragic way from the fellow that was known during their life. Everything is there, but crudely rearranged. They are not the same, yet it is still them. Even in their decomposition they can be identified. Without doubt. It is the field of homeomorphism, the variability within unity.
Here the field of situological experience is divided into two opposed tendencies, the ludic tendency and the analytical tendency. The tendency of art, spinn and the game, and that of science and its techniques. The creation of variables within a unity, and the search for unity amongst the variations. It can be clearly seen that our assassin has chosen the first way, and that the identifiers must take up the second, which limits the domain to the analysis of sites, or topology. Situology, in its development, gives a decisive push to the two tendencies. For example, take the network represented by Galton's apparatus. As a pinball machine, it can be found in most of Paris bistros; and as the possibility of calculated variability, it is the model of all the telephone networks.
But this is the creative side, which precedes the analytic side in general and elementary situology: the situationists are the crushers of all existing conditions. Thus we are going to start our demonstration by returning to the method of our criminal. But to avoid making this affair a bloody drama, we shall dive head long into a perfectly imaginary and abstract world, like Euclid.
We start by lending an object a perfect homeomorphism, an absolute and practically nonexistent quality, like the absence of spatial extension that Euclid gives to his point. We give absolute plasticity to a perfectly spherical ball with a precise diameter. It can be deformed in any way without being broken or punctured. Our goal is clear before this object of perfect three-dimensional symmetry. We are going to completely flatten it to transform it into a surface with two dimensions and to find the key to their homeomorphic equivalence. We are going to reduce the height of this sphere to zero in ten equal stages, and calculate the level of increase of the two corresponding dimensions to the registered reductions of the third progressively as the ball is transformed more and more into a surface. The last number can be deduced from the preceding nine. It is evident that we don't end up at infinity, as the same process with a ball five times as large must give a surface at least five times as big, and two infinities with a difference of measurable dimensions is beyond logic (except for Lemaître when he speaks of eternity). The practical work of calculation linked to this experiment, we shall leave to the mathematicians - if they have nothing better to do.
We haven't finished. We choose a diagonal in this immense pancake without thickness, and start to lengthen the surface in exactly the same way as in the previous experiment, to end up with a line without thickness, making the calculations in a similar fashion. Thus we have the homeomorphic equivalence expressed as numbers between an object in three, two and one dimension, and the whole world can start to protest. The most intelligent will be patient, saying that Euclid started with a point. How is this immense line reduced to a single point? I can only return to the sphere. If the situology was a uniquely spatial and positional phenomenon this will be true.
Einstein has explained that if a line can reach the speed of light, it will contract until it disappears completely as regards the length along the direction of the trip. However a clock would stop all together at that speed. This is what we are going to do. The whole matter is settled in this way. The only minor inconvenience of this spectacular process is invisible: I cannot regain possession of my point, which flies off across the universe. If I could transform this movement across space into rotation in place, I would have more or less mastered my point.
Einstein declared the "space and time conceived separately have become empty shadows, and only the combination of both expresses reality". It is from this observation that I'm going to clarify the Euclidean point, which possesses no dimensions and, as it is within space, before however representing any other dimension, at least represents the dimension of time introduced into space. And it is all the more impossible to fix a point without duration in space. Without duration there is no position.
But in order that this point can possess the quality of time, it must possess the quality of movement, and as the geometric point cannot be displaced in space without making a line, this movement must be rotational, or spinning around itself. Although this movement must be continued, it does not however have an axis nor spatial direction; and what's more vortex cannot occupy the least space. If this definition of the point is richer and more positive than that of Euclid, it does not seem to be less abstract. But since I have learnt that there is a Greek geometer, Héron, who inspired Gauss with a definition of the straight line as a line which turns around itself as an axis without the displacement of any points which compose it; and that plenty of people agree that this is the only positive thing which has ever been said on the subject of the straight line, I feel I'm on the right track.
But an axis can only have a rotation in a sense. It is necessary to stop it to spin it in the contrary sense. However a point in rotation, by a continuous change of its axis of rotation, could be led to a rotation in the contrary sense, whatever the sense. In this way the straight line can be explained thus: If two points rotating at random are connected, they are obliged to spin in the same sense and with the same speed, the faster being braked and the slower accelerated.
All the points of a line acquire a presence in the spatial dimension equivalent to their loss of freedom of movement, which has become oriented in space.
If we want to stay with this oriented and positive definition of the line on our backs, a plastic definition is needed. To understand this, it is necessary to remind ourselves that plastic geometry does not place the accent on the infinite character of dimensions, but on their character of a presence in general space and time, which could be finite or infinite, but which are primarily in relation with all the objects whose extension is wanted to be studied. Each volume, each surface, each segment of line or piece of time makes a part, or is extracted from the general mass of universal space and time. In the analysis, for example, of a linear segment in the egalitarian geometry of Euclid, abstractions of an 'infinite' character are made of the line. A piece is cut away by forgetting the rest. In unitary geometry, this is not possible. A line is not an interrupted series of points, because the points have lost something in order to be able to establish a line. In a segment of a line, there are only two points which could be observed, the two points at each end of the line. But how is it explained that on a line segment there are two rather than a single zero point? The only possible explanation is that a line segment with two zero points is composed of two demi-lines superimposed, with the zero points crossed, going in opposite directions. A line segment is thus a line to double distances, there and back, and of a length double the distance between the two polarized ends or in counterpoint. This is a basis for plastic or dialectic geometry. From this outlook, each determined volume is a volume within general volume, or universal space, fragmented by a surface: just as each surface is a fragment of a surface distinguished by some lines; and each linear section is a linear segment determined by its duration.
The specific surface which determines a volume, the voluminous surface is termed the vessel, form etc. And as a function of separation between two volumes it possesses the character of an opposition between the inside and outside; similarly the separation of a surface by a line opposes before and after, and so also the point on a line distinguishes the positive and negative sense of distance. These signs thus only make sense as the relation between two-dimensional systems, in the same combination of co-ordinates. The problem becomes more complex when we start to play with several co-ordinate systems in relation with each other such that it could be termed projective geometry, of which the best known example is central perspective.
In order better to understand not only the system of projections, but also the system of objectification in general it is necessary to see how the co-ordinate systems unfold and which is the initial primary system. The primary system of all observation is the system of co-ordinates inherent to the observer themselves, their subjective co-ordinates. Ordinarily this elementary requisite for observation is ignored. The co-ordinates of the individual are known as front, behind, above, below, left and right; and they play an enormous role for orientation, not only in science, but of a primordial way in ethics, the social orientation where the individual is drawn to the left and then the right, toppling forwards, always forward thanks to progress, pushed from behind and pressed towards the ascent and the higher pathways, to finally be carried underground. The direction to the right is the direction of least resistance, of the right line, the direction said to be just or rational; and opposed to it, the left is by nature the anarchic direction of the game, of the spinn or of the greatest effort. But each time that the political left becomes the direction of a development of justice, following the path of least resistance, this opposition lacks tension. The trajectory of descent is delineated by the path of least resistance. So, from our outlook of oppositions, the left direction of the left, that of games, must represent the ascent. This is what I have tried to prove with the reversal of dialectics. In the Scandinavian languages the word droite (German recht, English right) mean ascension (högre) towards the heights, which symbolizes the left elsewhere. The confusion in social orientation in Europe and in its vocabulary gains from being so rich and contradictory in this respect. These are purely objective observations, without any pragmatic consequence, but which have had an influence even on the most elementary religious conceptions (heaven - fire).
In reality the metric graduations of a co-ordinate system allow the establishment of a network of parallel lines of co-ordination at equal intervals. The zero point and the positive directions can be chosen and changed in the system as it is desired thanks to this squaring up. It is the same thing for the line and for the system of three co-ordinates.
When the system of co-ordinates of an observed object is displaced in relation to the basic system of co-ordination for observation and measure, this sometimes necessitates projection. The projective geometry thus shows the rules of the relations between two or several systems of co-ordination, as if there were two or several spaces. In this way, the same space can be multiplied into several by projection. But this is only justified through the time dimension.
However, positive geometry, which works with the demi-line, the quarter surface and the eighth of volume, allows another purely spatial game. The right angle formed by two negative demi-lines of a co-ordination in two dimensions can be displaced and put in opposition to the positive angle, thus establishing, for example, a square. This operation explains how the square could find its explanation in the relationship between the circumference and the diagonal of a circle, even though the circle cannot be defined as a derivative of the square. This definition of the square by juxtaposition joins our dialectic definition of the line, and shows how situology is more immediate than geometry, which always runs into the problem of squaring the circle.
Here we have roughly sketched out some consequences of the disorder, which situology could introduce to geometric thought, but it is evident to those who know this material, that the consequences will not be any the less as regards our physical and mechanical conceptions. It has already been understood by Einstein's definition that the notion we have of light doesn't lend itself to any spatial dimension. However it would be wrong to consider light as being immaterial. Even the old mystical notion of the four elements could be reconsidered. We know that they don't exist as absolute phenomena, but it is however strange that modern science has refused to consider a distinction of matter as pronounced as that between solid, liquid, gaseous objects and light. When an ice cube suddenly melts and stretches on the surface of a table, it can be concluded that the liquid state represents the loss of one of the spatial dimensions, replaced by the liberation of discharge; that the liquid is a matter of two spatial dimensions. And the constant of tensions of surface tension seems to be as important in physics as the constant of the speed of light. The logical conclusion this gives rise to, is that gases have only one dimension, compensated for by the play of their movement. And for an example of something, which has even less dimensions, think of Maurice Lemaître and his friends.
Translated by Fabian Tompsett. From https://www.cddc.vt.edu/sionline/si/open4.html