Logic will get you from A to B imagination will take you everywhere” Einstein.
The genuine mathematics –as Kurt Gödel showed — can be classified in abso-lutely no Formal Axiomatic System, no matter how advanced that system is.They cannot be formal; they are by default informal (non typical). The logicalassumption is then, that there cannot be a (formal) mathematical theory.Therefore, the nature of genuine mathematics is not objectively “mathemati-cal”. 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 princi-ple 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 pro-duced logically, because if it were so, they would not be axioms, they would betheorems of logic. So, mathematics is not a simple acceptance of logical prin-ciples and metamathematic rules. If there is no creation of new axioms, wherethere is a need of them, then mathematics stop developing. There is no re-striction on the choice of the axioms, except for the non-contradiction andtheir sufficiency. However, those two principles are not enough for the cre-ation of axioms. They are the negative terms that have to be abided for the ax-ioms to be valid and acceptable. The mathematics of thought exceed the totalsand deal with magmas. For example, all the pictures that we keep in ourminds when we are thinking or dreaming do not form a set in mathematicalterms. That is because a set consists of elements defined and distinguishedfrom each other, which come either from the real world or from the notionalimaginary world. Our pictures, however, do not consist of elements distin-guished and defined. We cannot separate them; all elements are connected toeach 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 whichexpress the earthly dimensional (Euclidean) space-time environment. Theseextreme spot symbols of mathematical expression are nothing else but the extreme sections of the net (intangible) mathematical universe.
Namely, theyare 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 writtenlanguage. Both the oral and the written language are nothing more than aform of the net structure of thought — logic. Logic in turn, should not be coincided with mathematics (neither as being a mathematical logic), becausemathematics 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 inempiricism (“sensualism”), or in intuitionism (“intuitive mathematics”) either. They refer directly to the queen of human capabilities in Communica-tion, i.e. imagination (and the Imaginary).“Imagination is more important than knowledge. For knowledge is limited toall 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 helacks is the “other”, the unseen and invisible world. Man imagines what helacks and he creates it with imagination. “Nothing exists in thought if manhas not imagined it” (Aristotle). Creativity, creative imagination (Kant) is theability to create new ideas, forms and situations from void and nothing.‘’Imagination is the queen of truth, and possibility is one of the regions oftruth. 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 ismore important than that thing itself. However, imagination does not onlycreate images. It also creates wishes and provocative excesses. Unquestion-ably, 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. Imagina-tion is required in order to surprise, to satisfy and to move them. Imaginationmeans communication and creation after all. Furthermore, communication isthe creation of the imaginary; it is an address to the imagination. This explains why communication is creative in more than one ways. Imagination,then, is necessary to Communication, because beside the creation of images, words, meanings, universe, wishes and dreams, it also results in offering thepleasure of knowledge or the knowledge of pleasure to the world… It offersthat is, mathematical imagination itself!
1.According to the special Theory of Relativity, the material universal realityis nothing more than the projection, the image, the reflection of everythingenclosed in the four dimensional non Euclidian and invisible universe. Thefourth dimension is the projection to the four dimensional and Euclidianspace that is created by the physiology of our senses.
2. The language must not be identified with the Thought nor must it be dis-tinguished from it, as the visible world of the perceptible material is not iden-tified nor is it distinguished from the invisible world of the super noticeableenergy. Time is the rhythm that combines the opposites without identifyingthem.