“Logic will get you from A to B imagination will take you everywhere” Einstein.
As Kurt Gödel showed, can be classified in absolutely no Formal Axiomatic System, no matter how advanced that system is. They cannot be formal; they are by default informal (non typical). The logical assumption is then, that there cannot be a (formal) mathematical theory. Therefore, the nature of genuine mathematics is not objectively “mathematical”. Namely, it is not governed by the logic of set theory. The logic of the set theory is governed by the principle of identity, the principle of non-contradiction, while the choice of their axioms is strictly poetical, that is “creative”. The choice of the axioms is not empirical, nor are they produced logically, because if it were so, they would not be axioms, they would be theorems of logic. So, mathematics is not a simple acceptance of logical principles and metamathematics rules. If there is no creation of new axioms, where there is a need of them, then mathematics stops developing. There is no restriction on the choice of the axioms, except for the non-contradiction and their sufficiency. However, those two principles are not enough for the creation of axioms. They are the negative terms that have to be abided for the axioms to be valid and acceptable. The mathematics of thought exceed the totals and deal with magmas. For example, all the pictures that we keep in our minds when we are thinking or dreaming do not form a set in mathematical terms. That is because a set consists of elements defined and distinguished from each other, which come either from the real world or from the notional imaginary world. Our pictures, however, do not consist of elements distinguished and defined. We cannot separate them; all elements are connected to each other and are defined mutually.
Only the linguistic expression of genuine mathematics is typical, since, as being a language, it is a human construction of symbolic/point patterns which express the earthly dimensional (Euclidean) space-time environment. These extreme spot symbols of mathematical expression are nothing else but the extreme sections of the net (intangible) mathematical universe. Namely, they are material thickenings that take place due to the function of the human brain (1). An example may analyze the above: as electromagnetism is material (electricity) and “power” (“something else”) altogether, the mathematical universe
is language (material) and thought (“something else” – “energy”) together. The language should not be coincided with thought (2). On the other hand, neither the thought should be coincided with the written language. Both the oral and the written language are nothing more than a form of the net structure of thought – logic. Logic in turn, should not be coincided with mathematics (neither as being a mathematical logic), because mathematics are an immaterial – invisible “language” which is expressed as a visible material through mathematical symbols – spots. Ultimately, Mathematics themselves cannot be categorized, as we showed, in rationalism or in empiricism (“sensualism”), or in intuitionism (“intuitive mathematics”) either. They refer directly to the queen of human capabilities in Communication, i.e. imagination (and the Imaginary). “Imagination is more important than knowledge. For knowledge is limited to all we now know and understand, while imagination embraces the entire world, and all there ever will be to know and understand.” Einstein. As Hegel said, desire is the matrix of Thought. Man, desires what he lacks and what he lacks is the “other”, the unseen and invisible world. Man imagines what he lacks and he creates it with imagination. “Nothing exists in thought if man has not imagined it” (Aristotle). Creativity, creative imagination (Kant) is the ability to create new ideas, forms and situations from void and nothing. ‘’Imagination is the queen of truth, and possibility is one of the regions of truth. She is positively akin to infinity’’. Charles Baudelaire
As Baudelaire said, imagination is the queen of the human capacities. Indeed, reality is unified with the imaginary and the image we create for something is more important than that thing itself. However, imagination does not only create images. It also creates wishes and provocative excesses. Unquestionably, both the greatest and the most awesome human accomplishments were born in his imagination in the first place. We need imagination in order to “touch” others and spread ideas strange or even annoying to them. Imagination is required in order to surprise, to satisfy and to move them. Imagination means communication and creation after all. Furthermore, communication is the creation of the imaginary; it is an address to the imagination. This explains why communication is creative in more than one way. Imagination, then, is necessary to Communication, because beside the creation of images, words, meanings, universe, wishes and dreams, it also results in offering the pleasure of knowledge or the knowledge of pleasure to the world… It offers that is, mathematical imagination itself!
- According to the special Theory of Relativity, the material universal reality is nothing more than the projection, the image, the reflection of everything enclosed in the four-dimensional non-Euclidian and invisible universe. The fourth dimension is the projection to the four dimensional and Euclidian space that is created by the physiology of our senses.
- The language must not be identified with the Thought nor must it be distinguished from it, as the visible world of the perceptible material is not identified nor is it distinguished from the invisible world of the super noticeable energy. Time is the rhythm that combines the opposites without identifying them.