There is an immutable present time that accompany the origin of Opera and its evolution, that delivers its horizon to eternity, but never assigning it a dimension out of time.

If we follow these lines, we step already towards its deeper origin which is to be found in the theatrical form of Tragedy as it was conceived in the ancient Greek world and how it sprouted from questions that looked towards an absolute comprehensive description of existence.

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Here we also foggily see, in this call for the absolute, the reverberances of the shores we intend to bridge together. Let´s still remind us that we departed from Opera and we are still looking over the water with lenses which are accordingly build to the realm we moved our first steps from. Sharpening the lenses along the research will allow us to reach a clearer description of the realm its expecting us on the other side.

I take in this direction the chance to enfold an understanding upon Tragedy driven from inspirational readings of literature, focusing on few relevant passages extracted from them and sharing intuitions and reflections, with the aim of giving more concreteness to the construction of the bridge.

 

From Chaos as unlimited space, untighten still from any relation to inherent order, where the most diverse forces come to interaction, the Greek world started first to literally invent and apply a map of orientation upon it called Mythos, a sense-making reaction to an extra-ordinary presence of Existence.

 

Already in this sentence many terms call for definition.

Some will be not yet touched. Some will be assumed as intuitively given or relatively to others.

Let's begin considering: unlimited, space, existence. 

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To jump from this location of the ancient Greek Universe, to the beginning of the 19th Century would give us more raffinate means to operate this extraction or assignment of meaning. To define them by this catapulting act over historical contexts is a very delicate and fragile strategy but making more sense in terms of Letter to the modern mind.

The suggestion is, instead of describing the considered words through the eyes of ancient Greeks, which can only become an exciting retrospective play of projecting over the past modern idealizations upon it, or through the lenses of modern logical standing points, more anchored in the present time, to seek for a fertile negotiation, between what these words can still evoke at our time, coming as fruits of significant deepening and iterations along history, and how they have been unlocked in the field of mathematics.

What I see coming to happen is the echoing between what mathematical insights in their preciseness of language and crystallin visualizations depict and the forces that motivated the mythological spaces to take their forms and blew meaning to their narrations.

Relating to the sight of the bridge, what´s coming to light is also the sharpening of the lenses to reach over the river´s side.

 

UNLIMITED,

ITERATIVELY

ANEW GENERATING

 

The question about the fundaments of mathematics accompanies the most significant developments of this science since any kind of computational elaboration was ever conceived.

By associating objects of daily use to marks and signs, with the intention of setting a distinction between them, an incredibly wide step in terms of abstraction was taken, within the most quotidian and pragmatic actions of organizing and keeping account.

The relevant configuration of asking, if and how accessible, or possibly formulated, the question of fundaments is, while operating in forms of numbers and their basic, practical relations, is finding its more approximate resolution along its evolution over centuries and only in the moment mathematics is deliberately and significantly reflecting upon itself, this configuration takes a critical demarcation. Operating on the elements that seemed to be the fundaments of mathematics was an achievement of mythological degrees, where a look towards a wider sense of existence is finding its orientation and coordinates for becoming a deliberate act.

In the primitive state, while defining differences between the knots on a rope and tightening them after each other to distinct separate days, these units for accounting time, from being the same, because all naturally following the cycles of light and darkness, yet different along the wider arch of continuously proceeding days, they are brought a newly together, as the same, in abstraction, now all of them identified as numbers, yet differently represented through symbols, so that the original objects are now existing as elements of a nature that only deals in terms of visualization and order. Differentiating and simultaneously unifying through the abstraction.

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What is given as so understandable by its use, might not be so comprehensible by itself.

As if the moment a functionality is defined, also an aspect of the nature is lighted.

It seems so immediate to know how to operate with numbers in their basic forms, that this efficiency creates usually around them a close circuit in terms of their definition, shifting constantly the attention towards their practical meanings.

To ask what the nature of numbers is, is leading abstraction capabilities to an extremely wider level, that for many centuries this question was even dis-considered by mathematicians themselves. 

In the description of the primitive habit of keeping track on ropes of past days, the words associating, continuous, following and order were used. All terms that have a correspondence in mathematical language. Intuitively and unacquainted with specific terminology, these words may evoke a sense of infinite. Referring to counting, the endless perspective, open ahead, while proceeding in terms of quantities, is revealed and ready to be subjected to any form of deformations in understanding, as long as it serves and finds its justification in practice.

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Reflecting on what the term infinite means mathematically, the question on the fundaments of this science started to be answered. This passage doesn´t seem to make the challenge easier when it comes to the nature of numbers.

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If it is so innate to be able to operate endless counting,

why it becomes so slippery to deal with infinite in its most natural form?

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When tools were created, also infinite could be elegantly and effortlessly domesticated. Basically, a machinery that explicates in mathematical terms the common meaning that we assign to the words associating, continuous, following and order, mentioned before.

 

SPACE

CONVERGENCE

FRAGMENTATIONS

 

Holding on this ride across unlimited, once the arithmetic scenario served as ground for entering a more primitive, yet already so abstract degree of infinte, now it takes a curve through the field of geometry, as a ramp to land back with more resolution on the constellation of numbers.

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By walking on a road, it is possible to count every taken step and describe a measurement of the distance covered along the way. This walk involves together with the perspective of the subject, the more objective one, which belongs to the landscape that gives ground and surrounds the walker. In a sense, according to the consistency of writing, it means to accompany Gauss on its measurements along mountains and rivers. A fictional way to proceed that can serve the purpose of defining space.

The distances covered can be related and represented in form of proportions. Only by using the natural numbers applied by our counting, we can embrace a more raffinate resolution of the wide scape of what are accounted as numbers, introducing the ones generated by any kind of undertaken or possibly foreseen proportion between the ones representing steps.

As a combinatory play, rolling two dies of infinite facets and reporting all possible couples of results. Each of these pairs, in terms of distance covered, will represent segments of our homogeneous cycling proceeding with steps of unitary size. 

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Still walking blindly immersed with the subjective point of view.

If the perspective of observation is shifted to the landscape and road, there is a sense of continuous and infinite that overcome all the most fractured and microscopical fragmentations revealed by the yet unlimited enfoldments of combinatory results.

How to embrace it from the subjective point of view?

With an element of finiteness and the structure of a convergence.

Leaving the idyllic scenario of hiking through valleys and mountains and displaying the etherical one of numbers, it is of a pivoting effect the decision of introducing a pre-term to limit, extremity.

When limit could lightly flee into limitless, extremity, by projecting far, still set a range of consideration. Each step is a measurement that partly, relatively describes  the distance and the unit of measurement. All possible segments of covered and possibly predicted distances, reduced to this unit, maintain the same density of occurrence. Therefore, focusing on the numerical interval between zero and one is a condensed representation of all possible incidents while rolling the naturally infinite dies. Extremities are set and the limitless sense is dressed by a more accessible vest.

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When the subject begins to think objectively, this question may arise: is it possible to represent by this structure of fragmentation any given minimal spot that the arch of the walker step could cover? May any infinitesimal small split of rain drop, that falls in between the traces left by the shoes, find its identifying coordinates on the system of segmentations?

Geometrically speaking, already if the zero-to-one segment, unit of the walking and simultaneously overall perspective, is bend together in the perfect form a circle and the radius is delineated, a proportion comes to emergence that flees all constructively predictable results of the combinatory segments. Indeed, this relation between the contour of the circle and the radius is reoccurring as the same without exceptions, no matter what is the size of the unit, and there fore deserves to be symbolized by the unique and outstanding sign π. 

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By not doing more than bending upon itself the unitary segment, the comprehensiveness of limitless proportions, fragmentations finds its innate contradiction. Only by setting extremities and tilting to convergence this overall mistakable comprehensiveness, a ground will be found upon which it is possible to walk more peacefully, with less indulgence.

 

EXISTENCE

SILENCE OF SEMANTIC

DANCE OF RELATIONS

 

If by Mythos we mean word, we can firstly understand it as an attempt to put order, as words do with letters and intend to do with the meanings they are accompanied by. A sense of order in terms of wordly narrated is generated, so that the relation between humans and universe, in which they find themselves as existing, is based on the tell of a story.

Existence by itself comes with order, in a state of Chaos, without meaning by it, impossibly readable in its manifestations, but intending the unlimited Whole, iteratively anew generated.

Humans felt the necessity to start, in their fashion, to pull threads that could give more resolution to the universal picture in its limitlessness, which they were to some degrees partaking and willing to act in, according to their needs of survival and therefore enabling manipulation upon their environment. 

Their intention was to assign meanings to the existence as they perceived it and reflected through the words of Mythos: a semantic invention. Words as carriers of an applied meaning.

Let´s consider an alternative to this craftmanship.

Instead of a semantic injection, a structural contemplation.

Meanings at side. Leaving only the forms of articulation of these words.

On the other side of the mirror, living from the forms of articulation of Chaos.

Extracting from Chaos to find forms of relation. With Chaos. Among its forms of becoming.

This is what is meant by existence as silence of semantic – dance of relations.

Mathematics will travel through this realm it in an accurate and depictive way.

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While Gauss was pulling the threads, letting traditional geometry of flat surfaces decline, as innate and only description of space, his procedures began to awake together with this revelation also the intuition that other models for understanding where existence takes place whispered their legitimacy to be taken in consideration.

The confrontation with the curved surface of Earth, from being and obstacle towards the demonstration of a celebrated perspective, became the leverage upon which to depart towards the so called non-Euclidean geometries. This non, firstly, only accommodates the sense of non-flat, welcoming the circle, or other geometrical construction from the Euclidean repertoire of shapes, become references and grounds for new spatial models; and also leaving the possibility open that not all or not only Euclidean assumptions are coming at play.

Nothing new was invented, only discovered and intended, adjusted to the well-known.

The set of tools and configurations, collected and developed by Euclid, designed properties of a dress that now asked for their capability to be assigned, worn by different bodies than a thin, flat one.

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Was it there, in this necessity and demand for flexibility of structures, the possibility to bend what was previously straight and sharply angled into, for example, a surface completely smooth and round, where edges disappears?

 

To bend only, meaning to use what was already laid as a consolidate architecture and to let the different body definitions, surfaces, measurements adjust their coordinates to cohere with the already designed. If, after this adaptation of the original model, a specular, tight one-on-one relation could be set between the previously settled and the secondarily re-defined and adjusted, then the legitimacy to exist and the validity of the contents for these new dresses for the body of space could be accepted. So it was.

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From this stand of things, an innumerable account of non-Euclidean geometries can be abstracted and grow from Euclid´s germs. His geometry, as mentioned at the beginning of this research, was always and only meant to describe plain figures. Gauss moved his steps into, through space; he immersed himself, with all is accounted knowledge upon straight measurements, into an ideally described scene, as three-dimensional, but only representable, also ideally, in two dimensions, just with the magic of perspective applied to it.

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Now the question left open previously becomes deciding and fundamental for all what was enfold related non-Euclidean configurations: is it the geometry of perspective in any relation to the geometry of plain surfaces? Yes, and as far it was with the ones generated from the Euclidean one, because as much as a sphere appears less practicable as a cube at enough large scale they practically work the same.

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Where to direct our eyes then, if the concrete body of geographies is not so determinant or essential anymore?

 

The one-on-one tight relation become the focus, as if matters would be held together by a very unmaterial thread that ground their existence, as a specular reflection, as identification under this relation could base the demonstration of existence.

One-on-one means exactly that both elements are consequence of each other, therefore they can be considered the same under the relation set between them, a mirror that let them identify mutually in each other. The logical term for being consequence is deduction.

This relation becomes the matter of mathematics. 

 

Plato´s overviewing perspective serves now to articulate and create more space within this entanglement. The principle upon which the definitions of philosophy and mathematics were presented in his Academia is related to the stand taken in front of the ground elements of these investigations and the consequential decision-making moment generated; apparently, a methodological aspect, more properly, touching their contents.

When mathematics was assuming undeniable bricks in order to construct building of meaning, and operating on from them, philosophy was departing towards more undoubtful elements that could give foundation to its original bricks. Both holding in their hand the leverage of intuition, one was assuming that a base was found and set, the other was not to be yet satisfied by the depth of the ground. Dia is a preposition from Greek language that accommodates together these diverging paths and unifies the directions originated by standing in front of, against the fundaments.

Against without any meaning of aversion or competitiveness, only adding to the standing point an attitude of iteration. This confrontation mode is capsuled in the dia form that set a coordinate of departure. Plato will call the philosophical method of seeking further in depth, dialectic, assigning much value to the scenario of opening a discussion and being able to investigate by spoken words, and will refer to the mathematical one of extracting, enfolding from set fundaments, dianoia, where noia, in the context set so far, is the intuitive capability noein, at its sharpest reaching towards the intelligible nous, where subject and object of intuition are meeting ontologically.

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