THE SIGN OF ADDITION

AND THE SYMBOLIC CONTENT IN

BOLYAI-GEOMETRY

Csaba HEGYI

Department of Painting, The Faculty of Music and Visual Arts, University of Pécs, Damjanich u. 30, H-7624 Pécs, Hungary, e-mail: hegyics@art.pte.hu

Received xx May 2011; accepted xx May 2011

Abstract: Original meaning of two words: the Greek ballein (βάλλω) and the Latin aleatory (aleatorius). The signs of subtraction (−),division (/), and absolute value (|a|). The meanings of the words hyperbola (ύπερβάλλω) and symbol (σύµβολον). The sign of the Greek and Saint Andrew's Cross (Crux decussata), addition (+) and multiplication (X).

Origin in the Eucledian geometry and in the hyperbolic space. The center point of the ball and the Cartesian coordinate system on the sphere. The crucifix: its horizontal direction is a stretching line, reciprocal vector (←→, ↔), its vertical direction (on the cross) gravitates (↓), and rises at the time of the church building (↕). Corpuscular matrix, the mosaic and the experience of color in divisionism (the centre point of the quadruple division, grid and pixel). The modern gnostic idea of the geometrical model, the logic of “there is only one whole” and a defense of the immortality of the soul (Bernard Bolzano).

Keywords: symbolic meaning of operation signs, cross, mosaic

1. Etymology of some mathematical expressions

Original meanings of words indicate correlations. The Greek word for tumbling dice has not become part of western languages. The Latin aleatoria has become an important term of modern music. However tumbling dice (rolling the dice) remained the symbol of chance. The word symbol in its original meaning is linked to chance. The word ballein together with all its prefixes helps to understand the hidden personal element in mathematics.

For examples: (σύµβολον [sumbolon, “a sign by which one infers something; a mark, token, badge, ticket, tally, check, a signal, watchword, outward sign”] < συµβάλλω [sumballō, “I throw together, dash together, compare, correspond, tally,

come to a conclusion”] < σύν [sun, “with, together”] + βάλλω [ballō, “I throw, put”], additional examples: µεταβολή [metabolē, “change”], from µετά [“meta-”] + βάλλω [ballō, “I throw”]; διαβάλλω [diaballo, “I slander”], from διά [dia, “across”] and

βάλλω [ballo, “I throw”]; ἀνάβάλλω from ἀνά [ana, “up” ] + βάλλω [ballō, “I throw”]). The personal element in mathematics is the background or basis of the mathematician's personal impetus, and that is which always renews thinking. Realizing does not always have a sign that can be foreseen, realizing is not always hypothetical. We lose the personal source if we deprive the mathematical symbols of their original nature. Ferdinand Ebner holds that besides trying to absolutize the “I” in the word of mathéma, mathesis, the relation to “You” is also supposed at the same time [1]. The sign of extraction instructs us to an operation. a-b=c, that is “a” will be less with what disappears from the result. The land and the sky get separated from each other on the horizon, on the skyline of the sea. Finally we will always reach the coast if we come round any mountain. We can see the largest horizontal straight line on the sea which is part of the concurrence line of an irregular ball. This “straight line” is the most aspectable as is the dimmest star the most visible point. Extraction leads to an actual state among natural numbers. The nothing (the zero) cannot be considered a natural number if we consider numbers natural.

The absolute value of the number written between the vertical axes, this number can be only positive. This number is saved from being a subtractor in itself. The operation of addition unites two entities detached so far. The sign of extraction involves in itself the sign of addition. Addition means growing. Time is growing, increasing and decreasing simultaneously. The vertical axis reminds of the living present. Béla Tábor writes about the symbol in the epilogue of his book (The Two Paths of Jewry):

“Symbols can be described by symbols only. The more desymbolised

the signs we try to use the more we fail in the description. The symbol is a sign, but a sign that hides as well as signifying; a chain of hidden layers of meaning each of which chases beyond itself towards a more hidden layer; and a more universal one. The symbol is a sign hiding an infinite chain of hiding signs. Being a sign, it signifies; but being a symbol it signifies also that it hides: that it hides this infinite chain of hiding signs. So the symbol does not signify less, but more than a conventional sign: it signifies also that it has deeper layers and these layers too, consist of hiding signs. Its hiding signifies, and its signifying hides – and both call.” [2]

2. The symbol of the cross

The cross is an ancient symbol. The sign of the extraction is not a symbol as is not the vertical dash either. In the cross turned to its side (Saint Andrew's Cross, Crux decussate), division and multiplication show similar relation to the previous one. The sign of multiplication indicates change. The sign of addition indicates finitesimal. The point does not have extension. The point of concurrence of addition and multiplication in Eukledian space is more lifeless, that is, they have weaker connection to coincidence. The etymological connections of the word symbol can be demonstrated by the mathematical conceptions of parabola, hyperbola and parameter. The symbol παραβολή (parabolē), from παραβάλλω (paraballō, “I set side by side”) <

παρά (para, “beside”) + βάλλω (ballō, “I throw”); ὐπερβάλλω ["I go beyond, exceed"] <

ὐπέρ ["above"] + βάλλω ["I throw"]; παράµέτρον from παρά [“beside”] + µέτρον [“measure”]. If we use the coordinate system only for the depiction of absolute values on the horizontal axis, symbolizing time, past and “future perfect” (future time up to fate) get equated. So the origin is always on the midway. Thus the vertical axis converges with the principal tenet of the hermetic tradition.

This is the same as the reciprocal vector interpretation of the cross. S.t. John of the Cross represents the crucifix from bird's eye view, from an inexplicable visual angle (some 450 years ago).

Fig. 1. A drawing of the crucified Christ

made by S. t. John of the Cross following a mystical vision [3].

Resurrection is erecting the church. I think so maybe the transfiguration is apart from this. The experience of color in divisionism uses invisible to make something visible. The wall joint of the mosaic that makes the picture a continuous one is also a division. The point does not have extension. The minimum distance is just a limit value. Here the Bolzano-Weierstrass theorem appears as an existential factor. The theorem states that each bounded sequence in Rn has a convergent subsequence. The space of Bolyai-geometry is not to be understood as an ultra-supernaturalist sign. We save dualism, the character present in neognosticism if we deprive the notion of hyperbolic space from its personal element. The idea will not be dualistic if the contact of the personal and of the mathéma, mathesis remains. Bernard Bolzano dedicates his book titled Athanasia, - a defense of the

immortality of the soul to his friend Joseph Hoffmann. This is an monadological essay, to console Anna Hoffmann when her fifth and last child died [4]. T. Molnar write:

„I fin din J. F. Donceel’s book, Natural Theology, a short sentence specifying, he says, what we mean by God. God is a personal being f infinite perfection, who has created everything from nothing, and who is distinct from the universe he has created [5]. (…) To the materialist, it opposes God’s personalness and his creation f the world (including matter) ex nihilo. To the spiritual pantheist it answers that God is personal yet not one with the world and its’spirit’, since he is distinct from universe. Against the humanists it argues for God’s infinite perfection (…), and, at any rate, for the existence of being who, although not human, it also a person. Finally, the definition refutes the ultra-supernaturalists when it confronts him with God’s attributes (…) which the ’god above god’ is not supposed to have, or at the very minimum, is not supposed to have in way that is positive and utterable.” [6]

3. The quadruple division

Lastly, I would speak about the quadruple division, the centre of mosaic and the pixel picture. The circle which can be drawn into the intersections of the wall joints of the mosaic covers a body part on the figural mosaic. In the mosaic the geometric mean of the pieces like circle centers bring out isolation. The icon does not cast a shadow (because the is nowhere to), there is no sculpture in the orthodox church because it casts shadow.

Fig. 2. Growing by way of the division of the space / decreasing by way of the multiplication of the space

Fig. 3. The expansions

The operation as an action gets equated in the corpuscular matrix. The drawing of the similarity between the beginning and the end can be displayed (see the operation curves). A Hungarian folksong from Transylvania says (unknown translator, detail):

O, love love accursed torture

why did you not blossom on every treetop?

On the top of every tree, on the leaf of a walnut tree,

so every maiden and unmarried young man would have plucked it.

Fig. 5. Illuminated leaf of a walnut tree

The flower of the love should bloosom on top of all tree, indeed, on the leaf of the big and strong walnut tree. On the leaf of a nut tree, which turns in the direction of light as a single-layered corpuscular matrix, we can see a mosaic structure. The leaf itself is a multitude, and even in itself is a mosaic. Neither the beginning nor the end can perfectly be symmetric.

4. Conclusion

The mathematics as a pure science may be strange to many people. We may dissolve this strangeness if we know that mathematics in all its aspects has deep connections with all spheres of the culture. The mathematician's person remains as personal one in the mathematics. Acknowledgements This work has been undertaken as a part of a project founded by the Pollack Mihály Faculty of Engineering, University of Pécs.

References [1] H. J. Green, F. Ebner, The word and the spiritual realities: a translation of and critical

introduction to Ferdinand Ebner's "Das Wort und die geistigen Realitäten" and a

comparison with Martin Buber's "Ich und du", Chicago, Northwestern University, 1980. [2] In. //home.fazekas.hu/~lsuranyi/Zsutoszo_angol.htm, in 1990 (translated by Pál Hegedüs

and Jessica Sacks). [3] In. G. Brenan, St John of the Cross: his life and poetry, London, Cambridge University

Press, Archive, 1975. The illustration is a reproduction of St John’s drawing of ‘Christ, crucified’, preserved at the convent of the Encarnación, Avila.

[4] Dov M. Gabbay, J. Woods, Handbook of the history of logic: The rise of modern logic: from Leibniz to Frege, Automatic control of converter-fed drives, Oxford, Gulf Professional Publishing, 2004, pp. 178-179.

[5] J. F. Donceel, Natural theology, New York, Sheed and Ward, 1962, pp. 13. [6] T. Molnar, Theists and atheists: a typology of non-belief, The Hague, Mouton Publishers,

1980, pp. 181-182.