Post by johnbc on Aug 16, 2020 23:47:47 GMT
The point is the traditional symbol of Being, or Unity. The simplest and most fundamental of symbols has been the one on which the greatest number of mistakes and paradoxes has accumulated, both in elementary geometry and in the psychological study of symbolism.
Of these paradoxes, the most surprising is the one that, having affirmed that the point has no dimension, declares that the lines and planes, like all geometric figures, are composed of points. How could anything be composed of something that, having no dimension, could be added to itself indefinitely without ever exceeding the zero dimension?
School geometry escapes this problem through the decree that point, line and plane are “intuitive” notions, implying therefore something like the popular notion about the “mysteries” of the Church, which although we cannot understand in any way we must accept willingly (as if it were possible to accept — or reject — a sentence whose meaning we do not entirely know).
In the case of Catholic mysteries, however, dogma leaves the door open to another form of understanding, stating that through faith and grace we will be able to assimilate a food that reason has in the account of indigestible; while the geometer assumes reason as the end point, recognizing no form of intellectual intuition as superior to this, and leaving no other option but to accept paralogism as the basis of logic and madness as the foundation of reason.
The irritated haste with which the geometry teacher slips over this point, repressing as impertinent the student who wishes to deepen it, is an invitation to the premature stupefaction of intelligence, which will be driven to be insensitized by daily coexistence with the mystery rendered harmless. We all know that school prizes go to those who do best in the skillful handling of mechanisms whose meaning they are totally unaware of and who will, in fact, maneuver with greater development and proud sufficiency the less they suspect the existence of a meaning, as this suspicion could bring it returns to the question of the fundamentals of reason, and to end in the paralyzing terror of the mysterium that extends beyond the utilitarian and self-complacent operationalism in which the “mathematics” of modern teaching are summed up.
Either we recognize that all essential knowledge is intuitive and immediate, with science being only the deductive application of intuitive principles to particular cases — with which we return to the medieval concept of science as art, or “application of doctrine” — or we accept that all science rests on an initial absurdity, to which it must periodically return, after a brief tour of particular phenomena and sensitive experience, to be devoured in the labyrinth and darkness as the ritual victims of a new Minotaur.
By diverting the students’ attention to the purely operational aspect — technical or pyrotechnical — of mathematics, and turning their ears to the appeal of its primordial, metaphysical and symbolic aspect, modern education becomes a self-indulgent and irresponsible dance on the abyss, preparing young people to get drunk later in the curious mixture of rationalistic pride and dark despair, which constitutes all the characteristic stench of modern cultural life.
The ancients, on the contrary, never failed to recognize that reason has its foundation and root in an intuitive form of knowledge, not, however, understood as a vague and indigestible obscure and infra-rational “mystery”, which we pass through quickly and in fear, like a thief in the night, to steal some axioms and run towards the technical and practical applications that constitute for us, today, the only clear and safe domain where we take shelter; intuitive form of knowledge understood, I mean, not as a blackness of the incomprehensible, but as a clear sky of contemplation (contemplatio, theoréin); beatitude of knowledge that was the ultimate goal of all pedagogy, all technique, all science, all rationality.
For the ancients, the “first principles”, known by intuition, were the origin, but also the goal of knowledge; but not in a circular, self-repetitive and dark process as in the case of the modern Minotaur, since the origin, the path and the end occurred in different planes.
The “origin” designated not only the logical or temporal beginning, but, on the contrary, the supra-temporal plane of archetypes or eternal possibilities
The path was, on the one hand, temporal existence and, on the other, the reason as a guiding thread or a map of return to the world of archetypes. Philosophy — science as such — was an activity aimed at correcting the deviations of the human mind, shaping it by the infallible certainty of archetypes, symbolized in numbers, musical harmonies, geometric figures and planetary spheres. Reason, therefore, led man to the portal of mystery.
But this mystery differed profoundly from the abyss of perplexity that is the starting and ending point of modern logic and mathematics
First, if reason was not the end point, but only the means or the path to lead to something else, the ancient philosopher would have no more reason to be frightened when he reached the border of the non-rational than would have been an traveler who, having taken a train to go to a certain city, saw the end of the journey approaching. Far from seeing this perspective as the end of the world, he would see it simply as the transition from the provisional to the definitive, from the middle to the end.
In fact, the very name of philosophy presupposes the existence of knowledge superior to philosophy itself, that is, of a terminal, definitive wisdom, “after the obtainment of which there is no more knowledge to be obtained”. Outside this hypothesis, it must be admitted that philosophers defined themselves from the beginning as lovers of the non-existent.
The passage from philosophy to wisdom is well marked in the structure of the Platonic dialogues, where the dialectical, — preparatory or properly philosophical — part always follows the mythical account, that is, the symbolic transmission of an effective and conclusive knowledge of a sapiential nature.
Second, the word “mystery” only very recently — from the Renaissance, as far as I know — came to mean the unintelligible. Before, it designated precisely something through which knowledge was revealed, became visible. If not, how can we explain that this word was used as the name of a theatrical, pedagogical and popular genre, such as medieval “mysteries”? Before that, however, the term mystery properly designated a phase of wisdom teaching — the “Little Mysteries” referring to the teaching of the laws of the cosmos and becoming, the “Great Mysteries” to the knowledge of God and eternity. When it comes to teaching, it is evident that neither the small nor the great Mysteries had anything “mysterious” in the current sense of the term.
Thirdly, the cyclical return to the mysteries did not have the aspect of endless repetition, in a closed circle that rightly could be considered an image of hell, because it was precisely a question of returning from manifest, and therefore finite, reality to the world of archetypes, and therefore of eternal possibilities, and from there to the Absolute, leaving definitely the whole cycle of transformations (samsara).
The return to principles thus had the function, on the one hand, to reassure the submission of the parties to a central and superior nucleus of principles and, on the other, to allow that central intuition to radiate again over the whole field of particular knowledge and applications, fertilizing it and renewing them.
Each return brought, therefore, a regeneration of the world, and, in this sense, the periodic return of science to its principles had a function analogous to the rites of renewal of time that all Traditions have always performed at the end and opening of each time cycle and of which the current year-end festivities represent a cartoon residue.
The point, it is said, is that which has no dimension or extension of any kind. Now, a dimension is nothing more than a system of directions that defines the various extensions according to which a figure admits to being measured. Depending on the minimum number of directions that define a figure, such will be its dimension. A line is defined by a single direction (two directions); one plan, for two; one solid, for three.
Euclidean geometry admits only these three dimensions, but we can use geometric, or spatial, symbolism to represent realities that are not in themselves spatial or geometric; for example, when we use the movement of a watch’s indicators to mark the time; in these cases, the geometric representation will imply more than three dimensions, although in the drawing they have to remain implicit, so to speak. It is clear that no symbolic system can account for the totality of reality, and that is why the ancients articulated several symbolisms to each other, by attaching, for example — in the quadrivium — music to geometry; in fact, a watch is a simultaneously geometric, musical and astronomical representation of time; and anyone can verify that the absence of any of these three representations would make the existence of this symbolic synthesis called clock impossible.
Any symbolic system is thus implicitly multidimensional, and geometry would have no way of escaping that, whether modern geometrists admit it or not.
Now, a point, if it has no extension, has, however, dimension, contrary to what is believed, because it has to be in some direction, under penalty of not being anywhere, that is, of not existing.
Well, in how many directions is a point? It is in all directions at the same time, because any line that you imagine, in any plane that is, will always have a parallel that necessarily passes through that point.
The point is thus the figure that, having no extension, is simultaneously in all directions and therefore has the totality of dimensions
In this sense, the point represents the logical and ontological principle from which the figures emerge, and not just a constitutive “element”; because an element, in order to contribute to the formation of the figure, should be added or articulated to other elements of the same genre, with which we would fall into the already mentioned nonsense, of the sum of inextensive elements ending up producing extension; whereas a formative principle necessarily contains the key to all the phenomena it produces, not needing to be added to whatever is of a different and superior reality to that where these phenomena occur.
Thus, having all directions and dimensions, the point also contains the formative key of all figures. These, therefore, cannot be formed by adding points, but, on the contrary, by suppressing the directions and dimensions of the point.
A line will thus be defined as one of the many directions that cross a point; one plan, like two; space, like three. The various directions and dimensions can thus be considered as points of view according to which the point can be focused; and geometric figures, as combinations and articulations of these points of view.
If a point, considered in itself, has all directions, considered as an “element” of a line it will have a single direction, precisely because of the unidirectional limitation that defines that line.
The dimensions and figures are, in this way, and so to speak, “subjective” in relation to the point, as they constitute only ways of looking at it, while the point is totally objective, therefore, containing in itself all points of view, does not depend on any of them to exist.
With that, we get rid of the pejoratively “abstract” character of geometry and restore its organic link with normal human perception, since, in the sensitive reality, we cannot “see” a point, except as intersection of lines, even so that we cannot “see” an object “in yes”, that is, in the simultaneity of all its dimensions, but only according to one or a few points of view, which will be precisely those for which we face it. The point’s invisibility is the invisibility of any object framed — and therefore limited — by a given system of perspectives. So that the apparent paradoxes about the point are found in any sensitive object, and it is not appropriate to attribute to the geometric objects a character that is neither more nor less “mysterious” than to all the others.
Thus, as the figures are formed by particularization — and therefore limitation — of the possibilities of the point, it is clear that the totality of the possible figures will be an integral manifestation of these properties and, therefore, the equivalent, in the order of manifestation, of what the point is in the order of principles.
This is symbolized in the relationship between center and circumference, since the circumference represents, in the plane, the same as the sphere in space. We know that the curve is determined by its tangents; the tangent, being a straight line, contains a direction (two directions). Therefore, the circumference, being the only figure that is defined by having an “infinite” (or rather, indefinite) number of tangents equidistant from the center, has, at its own level, one of the properties of the point, which is that of having a ‘“infinite” number of directions; the difference is that the circumference has “infinite” directions in the plane, while the point has them in space, being itself thus the principle of space.
As for the sphere, it has an indefinite number of tangent lines and planes in all directions, and could be considered totally equal to the point, if these tangents were also tangent to the center; now, the distance from the center to the tangent plane of the sphere — the radius — is not in itself a tangent to the sphere, and therefore the sphere has all possible directions minus the directions of the rays, and is therefore more limited than the point. There is a homologous relationship between the point and the sphere to that which exists, in metaphysics, between “Absolute” and “Totality”; the totality implies a quantitative consideration, (to which the Absolute is transcendent) and therefore, although representing the Absolute, it is not.
Of all the figures, the most similar to the point is therefore the sphere, because, the figures differing by their number of directions, both the point and the sphere have an indefinite number of directions. The same could be said of the circumference, in a flat symbolism.
The line, on the other hand, is the most different figure from the point, because it is the most limited in terms of the number of directions. The point and the line therefore form the two ends of a “scale” within which the various geometric figures are distributed according to the number of their directions. As the point, however, is not exactly a figure, but itself is the principle of the figures, it can be said that it is outside and above that scale and that therefore the first figure — the most multidirectional of the scale — is the sphere, thus being the sphere and the line the two extremes. In a decreasing sense, this scale would go from the sphere, through curved solids — topological surfaces — to regular polyhedra, from these to flat figures and from these to line segments and lines, more or less like this:
Scale of figures
1st Sphere
2nd Solids of curved surfaces with tangent planes not equidistant from the center.
3rd Polyhedra with n sides
4th Polyhedron with n-1 sides
5th Polyhedra with n-2, n-3 … sides
6th + 1 Flat figures with curves
7th + 2 Regular flat figures with n sides
8th + 3,4,5, … n Flat figures with n-1, n-2, n-3 … sides
9th + n Line segments
10th + Line
This scale is the symbol of the totality of the states of being, according to its progressive “removal” from the pure Being. The line symbolizes the principle of division — the substance — and the point of the principle.
An indefinite number of lines can pass through a point. Each segment of these lines has a type of double and simultaneous reality: it can be seen as part of a line or as part of the total plane, which is the emanation of the point and in which the line to which this segment belongs is but a point of view between many. In the same way, each entity can be seen either as a member of its own species, or simply as an entity, that is, as something existing.
Now, given a straight line and, in it, a segment, this segment cannot be measured — compared — with a segment of another straight line unless we suppose the existence of a common plane to both.
(Here it is necessary to open parentheses to explain that two parallels could not, by themselves, determine a plane, because either there is a distance between them, or there is not; in the latter case, both are the same line, and a single line does not determine a plane ; in the previous case, it is necessary to assume between them an indefinite number of straight line segments of equal length, perpendicular to both, and thus it is not just two lines that determine the plane, but it plus at least one more segment. Thus, two lines determine a plane as long as they are not parallel).
Now, if we talk about plane, we immediately report to the point of origin and crossing of the lines. Thus, the measurement — comparison — of segments presupposes the existence of the plane and the reference of all lines to the point, that is, of all those related to an Absolute.
Thus, each segment belongs, simultaneously, 1st to the point that originates the line to which it belongs; 2nd to this line; 3rd to the total plan; 4th to each of the straight lines that cross the point and spread across the plane, because, if the segment belongs to the totality of the plane, it also belongs to each of its parts, since these have nothing but the totality.
Therefore, we have a symbol of the simultaneous participation of the entities in various states of existence (represented, in this case, by the directions).
The entity participates in its own state through the direction it is in and, therefore, the distinction between that direction and the others; this distinction is made from the point. But it participates in the whole through the union of all directions at the point. And it also participates in each of the other directions through geometric figures that establish relationships between the various segments.
The symbolism of the circle and the line contains, in summary, all the cosmology. As the symbol of the Absolute, the point evidently represents the essential side, and the circumference, as a symbol of totality, the substantial side of the manifestation in particular, the point — representing the totality of possibilities, will pass, quite naturally — through the inversion that always occurs in the change of plane — to represent the substance of which the figures are made, and the line the essence, that is, what determines the particular quality of these figures. In fact, this is clear from the fact that, on the one hand, what defines the figures, when rectilinear is the direction and number of their edges, and, when curvilinear, the direction of their tangents; in both cases, they are lines or line segments that will determine the shape — that is, the nature, quality or essence — of the figures. On the other hand, since figures are nothing more than “points of view” about the point, as we have seen, it is clear that the figures are made from the point, from the point that is its substance, being more accurate to say that than to affirm, in the plural, as is generally done, that they are made “of dots”, which, in addition to leading to the contradiction that we have already noted, contradicts the unity of the substance on the cosmological plane.
So it is that, again by inversion, the rectilinear and regular figures will serve as a symbol of the incarnate Logos — the cross — while the circle will be the symbol of the transcendent Logos. Christ in human form is crucified; dead and transfigured, the “Sun of Justice” is in heaven.
For the same reason, rectilinear and regular symbolism will evoke the essential side of nature, its “divine” or heavenly aspect — for example, the three-dimensional cross that evokes Universal Man from the directions of space — and the curvilinear and irregular symbolism its substantial, “descending” aspect.
Of these paradoxes, the most surprising is the one that, having affirmed that the point has no dimension, declares that the lines and planes, like all geometric figures, are composed of points. How could anything be composed of something that, having no dimension, could be added to itself indefinitely without ever exceeding the zero dimension?
School geometry escapes this problem through the decree that point, line and plane are “intuitive” notions, implying therefore something like the popular notion about the “mysteries” of the Church, which although we cannot understand in any way we must accept willingly (as if it were possible to accept — or reject — a sentence whose meaning we do not entirely know).
In the case of Catholic mysteries, however, dogma leaves the door open to another form of understanding, stating that through faith and grace we will be able to assimilate a food that reason has in the account of indigestible; while the geometer assumes reason as the end point, recognizing no form of intellectual intuition as superior to this, and leaving no other option but to accept paralogism as the basis of logic and madness as the foundation of reason.
The irritated haste with which the geometry teacher slips over this point, repressing as impertinent the student who wishes to deepen it, is an invitation to the premature stupefaction of intelligence, which will be driven to be insensitized by daily coexistence with the mystery rendered harmless. We all know that school prizes go to those who do best in the skillful handling of mechanisms whose meaning they are totally unaware of and who will, in fact, maneuver with greater development and proud sufficiency the less they suspect the existence of a meaning, as this suspicion could bring it returns to the question of the fundamentals of reason, and to end in the paralyzing terror of the mysterium that extends beyond the utilitarian and self-complacent operationalism in which the “mathematics” of modern teaching are summed up.
Either we recognize that all essential knowledge is intuitive and immediate, with science being only the deductive application of intuitive principles to particular cases — with which we return to the medieval concept of science as art, or “application of doctrine” — or we accept that all science rests on an initial absurdity, to which it must periodically return, after a brief tour of particular phenomena and sensitive experience, to be devoured in the labyrinth and darkness as the ritual victims of a new Minotaur.
By diverting the students’ attention to the purely operational aspect — technical or pyrotechnical — of mathematics, and turning their ears to the appeal of its primordial, metaphysical and symbolic aspect, modern education becomes a self-indulgent and irresponsible dance on the abyss, preparing young people to get drunk later in the curious mixture of rationalistic pride and dark despair, which constitutes all the characteristic stench of modern cultural life.
The ancients, on the contrary, never failed to recognize that reason has its foundation and root in an intuitive form of knowledge, not, however, understood as a vague and indigestible obscure and infra-rational “mystery”, which we pass through quickly and in fear, like a thief in the night, to steal some axioms and run towards the technical and practical applications that constitute for us, today, the only clear and safe domain where we take shelter; intuitive form of knowledge understood, I mean, not as a blackness of the incomprehensible, but as a clear sky of contemplation (contemplatio, theoréin); beatitude of knowledge that was the ultimate goal of all pedagogy, all technique, all science, all rationality.
For the ancients, the “first principles”, known by intuition, were the origin, but also the goal of knowledge; but not in a circular, self-repetitive and dark process as in the case of the modern Minotaur, since the origin, the path and the end occurred in different planes.
The “origin” designated not only the logical or temporal beginning, but, on the contrary, the supra-temporal plane of archetypes or eternal possibilities
The path was, on the one hand, temporal existence and, on the other, the reason as a guiding thread or a map of return to the world of archetypes. Philosophy — science as such — was an activity aimed at correcting the deviations of the human mind, shaping it by the infallible certainty of archetypes, symbolized in numbers, musical harmonies, geometric figures and planetary spheres. Reason, therefore, led man to the portal of mystery.
But this mystery differed profoundly from the abyss of perplexity that is the starting and ending point of modern logic and mathematics
First, if reason was not the end point, but only the means or the path to lead to something else, the ancient philosopher would have no more reason to be frightened when he reached the border of the non-rational than would have been an traveler who, having taken a train to go to a certain city, saw the end of the journey approaching. Far from seeing this perspective as the end of the world, he would see it simply as the transition from the provisional to the definitive, from the middle to the end.
In fact, the very name of philosophy presupposes the existence of knowledge superior to philosophy itself, that is, of a terminal, definitive wisdom, “after the obtainment of which there is no more knowledge to be obtained”. Outside this hypothesis, it must be admitted that philosophers defined themselves from the beginning as lovers of the non-existent.
The passage from philosophy to wisdom is well marked in the structure of the Platonic dialogues, where the dialectical, — preparatory or properly philosophical — part always follows the mythical account, that is, the symbolic transmission of an effective and conclusive knowledge of a sapiential nature.
Second, the word “mystery” only very recently — from the Renaissance, as far as I know — came to mean the unintelligible. Before, it designated precisely something through which knowledge was revealed, became visible. If not, how can we explain that this word was used as the name of a theatrical, pedagogical and popular genre, such as medieval “mysteries”? Before that, however, the term mystery properly designated a phase of wisdom teaching — the “Little Mysteries” referring to the teaching of the laws of the cosmos and becoming, the “Great Mysteries” to the knowledge of God and eternity. When it comes to teaching, it is evident that neither the small nor the great Mysteries had anything “mysterious” in the current sense of the term.
Thirdly, the cyclical return to the mysteries did not have the aspect of endless repetition, in a closed circle that rightly could be considered an image of hell, because it was precisely a question of returning from manifest, and therefore finite, reality to the world of archetypes, and therefore of eternal possibilities, and from there to the Absolute, leaving definitely the whole cycle of transformations (samsara).
The return to principles thus had the function, on the one hand, to reassure the submission of the parties to a central and superior nucleus of principles and, on the other, to allow that central intuition to radiate again over the whole field of particular knowledge and applications, fertilizing it and renewing them.
Each return brought, therefore, a regeneration of the world, and, in this sense, the periodic return of science to its principles had a function analogous to the rites of renewal of time that all Traditions have always performed at the end and opening of each time cycle and of which the current year-end festivities represent a cartoon residue.
The point, it is said, is that which has no dimension or extension of any kind. Now, a dimension is nothing more than a system of directions that defines the various extensions according to which a figure admits to being measured. Depending on the minimum number of directions that define a figure, such will be its dimension. A line is defined by a single direction (two directions); one plan, for two; one solid, for three.
Euclidean geometry admits only these three dimensions, but we can use geometric, or spatial, symbolism to represent realities that are not in themselves spatial or geometric; for example, when we use the movement of a watch’s indicators to mark the time; in these cases, the geometric representation will imply more than three dimensions, although in the drawing they have to remain implicit, so to speak. It is clear that no symbolic system can account for the totality of reality, and that is why the ancients articulated several symbolisms to each other, by attaching, for example — in the quadrivium — music to geometry; in fact, a watch is a simultaneously geometric, musical and astronomical representation of time; and anyone can verify that the absence of any of these three representations would make the existence of this symbolic synthesis called clock impossible.
Any symbolic system is thus implicitly multidimensional, and geometry would have no way of escaping that, whether modern geometrists admit it or not.
Now, a point, if it has no extension, has, however, dimension, contrary to what is believed, because it has to be in some direction, under penalty of not being anywhere, that is, of not existing.
Well, in how many directions is a point? It is in all directions at the same time, because any line that you imagine, in any plane that is, will always have a parallel that necessarily passes through that point.
The point is thus the figure that, having no extension, is simultaneously in all directions and therefore has the totality of dimensions
In this sense, the point represents the logical and ontological principle from which the figures emerge, and not just a constitutive “element”; because an element, in order to contribute to the formation of the figure, should be added or articulated to other elements of the same genre, with which we would fall into the already mentioned nonsense, of the sum of inextensive elements ending up producing extension; whereas a formative principle necessarily contains the key to all the phenomena it produces, not needing to be added to whatever is of a different and superior reality to that where these phenomena occur.
Thus, having all directions and dimensions, the point also contains the formative key of all figures. These, therefore, cannot be formed by adding points, but, on the contrary, by suppressing the directions and dimensions of the point.
A line will thus be defined as one of the many directions that cross a point; one plan, like two; space, like three. The various directions and dimensions can thus be considered as points of view according to which the point can be focused; and geometric figures, as combinations and articulations of these points of view.
If a point, considered in itself, has all directions, considered as an “element” of a line it will have a single direction, precisely because of the unidirectional limitation that defines that line.
The dimensions and figures are, in this way, and so to speak, “subjective” in relation to the point, as they constitute only ways of looking at it, while the point is totally objective, therefore, containing in itself all points of view, does not depend on any of them to exist.
With that, we get rid of the pejoratively “abstract” character of geometry and restore its organic link with normal human perception, since, in the sensitive reality, we cannot “see” a point, except as intersection of lines, even so that we cannot “see” an object “in yes”, that is, in the simultaneity of all its dimensions, but only according to one or a few points of view, which will be precisely those for which we face it. The point’s invisibility is the invisibility of any object framed — and therefore limited — by a given system of perspectives. So that the apparent paradoxes about the point are found in any sensitive object, and it is not appropriate to attribute to the geometric objects a character that is neither more nor less “mysterious” than to all the others.
Thus, as the figures are formed by particularization — and therefore limitation — of the possibilities of the point, it is clear that the totality of the possible figures will be an integral manifestation of these properties and, therefore, the equivalent, in the order of manifestation, of what the point is in the order of principles.
This is symbolized in the relationship between center and circumference, since the circumference represents, in the plane, the same as the sphere in space. We know that the curve is determined by its tangents; the tangent, being a straight line, contains a direction (two directions). Therefore, the circumference, being the only figure that is defined by having an “infinite” (or rather, indefinite) number of tangents equidistant from the center, has, at its own level, one of the properties of the point, which is that of having a ‘“infinite” number of directions; the difference is that the circumference has “infinite” directions in the plane, while the point has them in space, being itself thus the principle of space.
As for the sphere, it has an indefinite number of tangent lines and planes in all directions, and could be considered totally equal to the point, if these tangents were also tangent to the center; now, the distance from the center to the tangent plane of the sphere — the radius — is not in itself a tangent to the sphere, and therefore the sphere has all possible directions minus the directions of the rays, and is therefore more limited than the point. There is a homologous relationship between the point and the sphere to that which exists, in metaphysics, between “Absolute” and “Totality”; the totality implies a quantitative consideration, (to which the Absolute is transcendent) and therefore, although representing the Absolute, it is not.
Of all the figures, the most similar to the point is therefore the sphere, because, the figures differing by their number of directions, both the point and the sphere have an indefinite number of directions. The same could be said of the circumference, in a flat symbolism.
The line, on the other hand, is the most different figure from the point, because it is the most limited in terms of the number of directions. The point and the line therefore form the two ends of a “scale” within which the various geometric figures are distributed according to the number of their directions. As the point, however, is not exactly a figure, but itself is the principle of the figures, it can be said that it is outside and above that scale and that therefore the first figure — the most multidirectional of the scale — is the sphere, thus being the sphere and the line the two extremes. In a decreasing sense, this scale would go from the sphere, through curved solids — topological surfaces — to regular polyhedra, from these to flat figures and from these to line segments and lines, more or less like this:
Scale of figures
1st Sphere
2nd Solids of curved surfaces with tangent planes not equidistant from the center.
3rd Polyhedra with n sides
4th Polyhedron with n-1 sides
5th Polyhedra with n-2, n-3 … sides
6th + 1 Flat figures with curves
7th + 2 Regular flat figures with n sides
8th + 3,4,5, … n Flat figures with n-1, n-2, n-3 … sides
9th + n Line segments
10th + Line
This scale is the symbol of the totality of the states of being, according to its progressive “removal” from the pure Being. The line symbolizes the principle of division — the substance — and the point of the principle.
An indefinite number of lines can pass through a point. Each segment of these lines has a type of double and simultaneous reality: it can be seen as part of a line or as part of the total plane, which is the emanation of the point and in which the line to which this segment belongs is but a point of view between many. In the same way, each entity can be seen either as a member of its own species, or simply as an entity, that is, as something existing.
Now, given a straight line and, in it, a segment, this segment cannot be measured — compared — with a segment of another straight line unless we suppose the existence of a common plane to both.
(Here it is necessary to open parentheses to explain that two parallels could not, by themselves, determine a plane, because either there is a distance between them, or there is not; in the latter case, both are the same line, and a single line does not determine a plane ; in the previous case, it is necessary to assume between them an indefinite number of straight line segments of equal length, perpendicular to both, and thus it is not just two lines that determine the plane, but it plus at least one more segment. Thus, two lines determine a plane as long as they are not parallel).
Now, if we talk about plane, we immediately report to the point of origin and crossing of the lines. Thus, the measurement — comparison — of segments presupposes the existence of the plane and the reference of all lines to the point, that is, of all those related to an Absolute.
Thus, each segment belongs, simultaneously, 1st to the point that originates the line to which it belongs; 2nd to this line; 3rd to the total plan; 4th to each of the straight lines that cross the point and spread across the plane, because, if the segment belongs to the totality of the plane, it also belongs to each of its parts, since these have nothing but the totality.
Therefore, we have a symbol of the simultaneous participation of the entities in various states of existence (represented, in this case, by the directions).
The entity participates in its own state through the direction it is in and, therefore, the distinction between that direction and the others; this distinction is made from the point. But it participates in the whole through the union of all directions at the point. And it also participates in each of the other directions through geometric figures that establish relationships between the various segments.
The symbolism of the circle and the line contains, in summary, all the cosmology. As the symbol of the Absolute, the point evidently represents the essential side, and the circumference, as a symbol of totality, the substantial side of the manifestation in particular, the point — representing the totality of possibilities, will pass, quite naturally — through the inversion that always occurs in the change of plane — to represent the substance of which the figures are made, and the line the essence, that is, what determines the particular quality of these figures. In fact, this is clear from the fact that, on the one hand, what defines the figures, when rectilinear is the direction and number of their edges, and, when curvilinear, the direction of their tangents; in both cases, they are lines or line segments that will determine the shape — that is, the nature, quality or essence — of the figures. On the other hand, since figures are nothing more than “points of view” about the point, as we have seen, it is clear that the figures are made from the point, from the point that is its substance, being more accurate to say that than to affirm, in the plural, as is generally done, that they are made “of dots”, which, in addition to leading to the contradiction that we have already noted, contradicts the unity of the substance on the cosmological plane.
So it is that, again by inversion, the rectilinear and regular figures will serve as a symbol of the incarnate Logos — the cross — while the circle will be the symbol of the transcendent Logos. Christ in human form is crucified; dead and transfigured, the “Sun of Justice” is in heaven.
For the same reason, rectilinear and regular symbolism will evoke the essential side of nature, its “divine” or heavenly aspect — for example, the three-dimensional cross that evokes Universal Man from the directions of space — and the curvilinear and irregular symbolism its substantial, “descending” aspect.
