RUPA (IMAGE), MATHEMATICS AND MATHEMATICAL ARTIST

1. RUPA (IMAGE), MATHEMATICS AND MATHEMATICAL ARTIST Mihir Kr. Chakraborty School of Cognitive Science Jadavpur University ， Kolkata, India

2. RUPA is a sanskrit word. Its immediate meaning is the visual form -- but it includes the beauty that emanates from the form • RUPA 是一个梵文词。其最直接的意思是视觉形态——事实上包含从“形式”中发出的所有华美。

4. Whatever might be the origin, if there be any at all, a mathematical object is ultimately a creation of human mind, an abstract entity. • Here is a line drawn on a piece of paper • 在纸上画一条线 • The mathematician looks at it as others do. But this real figure does not constitute her/his ‘object’. She now starts her (mental) journey, extends the length of the line in both directions and shrinks its breadth beyond any limit. Through this game of imagination, the actual line on the paper attains a new RUPA, a form or an appearance, not outside there on the paper but within the mind of the mathematical artist. This RUPA is what is called an Euclidean line. Like an art object, this is many times removed from reality.

5. Let us consider the creation of a projective line from an Euclidean line. • Let us add some more wishes to Euclidean lines: viz. • 让我们细思量一条欧几里得射线 * let there be a point called the point at infinity on each Euclidean line; * let any two Euclidean parallels have the same point at infinity; * let all the points at infinity constitute a line called the line at infinity ;One can notice that new imaginary objects are added with the existing ones. Also one can notice the role of language in the construction. • Thus is created the projective plane with projective lines and projective points inhabiting in it.

6. In this creation, there seems to be a heavy dependence upon the visual, a landscaping a la Wittgenstein but this visual is not only retinal, mind and words are equally and powerfully mingled in its construction.

7. Thus is born a mathematical object, a RUPA, an Euclidean line, a projective line or a transfinite number which is essentially abstract, a creation of the mathematician’s mind -- usually proposed by an individual and gradually coming into its full being through the sharing, interaction and contribution of the mathematician-community. • It should be cautioned here that there is no single homogeneous mathematician-community but many.

8. Word becomes indispensable. • (I do not subscribe to the intuitionist view-point viz. that mathematics is essentially a language-less activity.) Mathematical objects are constituted of descriptions. • Point: that has existence but no measure. • Number: that remains of a collection when all the attributes of individual objects of the collection are ignored by a ‘process of selective inattention’.

9. The mathematician passes from actual objects to imagination and then to words. Or from appearances of objects to words for a tentative and imagined object – words then reorganized to obtain a more satisfactory, yet imagined, object – the process being repeated several times until a RUPA emerges 直到一个RUPA涌现– that seems to be acceptable – that may or may not resemble the actual at all. Perhaps a very fine interplay takes place between the conceptual and the lingual in this weaving, this mental project, neither having primarity over the other.

10. Complex RUPAs are created out of the elementary ones e.g. topological group out of group and topological space. This combining act resembles again the act of an artist in that there is the need for arranging the component RUPAs in a harmony---in mathematical case the harmony being the compatibility conditions between the structures.

11. This new RUPA appears alien, mysterious and elusive to its creator. The creator has ascribed in it some initial properties and a few more properties might also be visible – but what else are there in it hidden ? Unraveling its mystery, then, becomes a compelling, unavoidable task to the mathematical artist. She enters into a dialogue with the mathematical object she has created just like a natural scientist enters into a dialogue with nature.

12. Is not an artist a discoverer too?

13. He creates but this creating is also a discovering. From the very first stroke on the blank canvas he enters into a dialogue with it. The canvas starts demanding, the artist may or may not satisfy the demands, may or may not be able to discover the ‘needs’ that are radiated from the canvas in its process of becoming an artistic creation. • The first stroke of the chisel on the blank marble starts the process of discovering the hidden shape within.

14. How beautiful or how potential a created mathematical object is may be measured by it capability of responding – it may stop to respond within years or it may continue over centuries. • 数学对象其美与深湛可以从其所蕴含及其回应来看——有时其回应会停止数年乃至延滞至数世纪。

15. A proof is a conversation between the mathematician and her created object – this object being ultimately laid down by a few axioms – a few initial claims about the object or a set of similar objects. In fact, these are the initial sentences that determine the object. Then other statements are derived by following reasoning methods and search techniques. It is by no means a trivial enterprise. One has to pose proper questions – find appropriate sub-questions (sub-goals) – make tremendous effort to obtain answers from the created objects about these sub-questions – and if obtained at all, one has to arrange them appropriately to get an answer to the basic question that has been posed. All these require very special language and signs.

16. A proof also is a creation of the mathematical artist. Although it ends up with a final claim, a statement about the RUPA arrival to which is the objective, the body of the proof itself fascinates. It is a RUPA itself. It means that the way itself of arriving at a conclusion interests the mathematician. We are not only concerned with whether the cat can scare the mouse, but also in whether it is black or white.

17. Proving mathematical statements is a very special kind of human activity and an outstanding product of human civilization. It is different from the common understanding of the word in NL and also from what is understood by it in physical and other sciences. What is obtained by a mathematical proof is infallible, timeless, once proved is true beyond doubt and to be accepted for ever -thus is the usual import of the concept. But the state of affair is not that linear.

18. A proof is a mathematical writing, a mathematical text that ‘establishes’ some ‘fact’ about some mathematical objects; the fact is expressed by a sentence that occupies the last position in the text. Once such a text is written it is thought that the claim in the last sentence of the text is established. In fact, so long as such a writing is not at hand with the sentence S at the end, one is not convinced that the claim made in S is established. S is never established by verification in ultimate analysis.

19. Usually a mathematical text is a mixture of NL-words, symbols and diagrams. As mathematics stands now, the entire text may be symbolized i.e. written as strings of signs. The converse question, namely can symbols and diagrams be dispensed with i.e. can any mathematical text be written in NL-words only – at least in principle?

20. I have deep doubt in this possibility. Although NL is involved in the creation of symbol, this involvement in most cases in just an initiation; • there are little chances of going back; mathematics proper can not take off so long as proper symbols (ways of writing in symbolic form) emerge with contents enough to endow them with rules of manipulation -- rewriting, checking mechanically etc. The actual flight begins with all its pleasure and ecstasy only after that stage.

21. A mathematician has two main roles to perform – as an artist, creating beautiful, potential structures, the RUPAs and as a discoverer, searching for the hidden, so far unrevealed properties of the created RUPA. The discovery is really generation of a dialogue.

22. There is a third role played by the mathematicians specially the instructors. This role is connected with another feature of a mathematical proof. • Interestingly, the adjective ‘elegant’ is often used before a mathematical proof. We may perhaps understand a ‘correct’ proof or an ‘incorrect’ one; but what is meant by ‘elegant’ in this scenario, how to judge this ‘elegance’? In this regard mathematics again bears similarity with an art object. To me a proof resembles more with a piece of music rather than any other artistic product. Creation of proof seems closer to creation of music. Firstly, from the structural point of view, both are sequential in nature – in one case series of sound-notes and in the other series of argumentation steps glued together.

23. Secondly, a piece of music not only gives pleasure to its creator but brings enjoyment to its listeners too. One may enter into a music repeatedly, a song is sung again and again. A proof when presented before a genuine audience brings extreme pleasure to its members if the proof is an elegant one. The same proof is presented by an instructor several times to his/her pupils – a creative instructor’s presentation each time is not the same, there occur interesting twists, re-organizations and small changes when a proof is offered to a group; each time it is an enjoyment to the instructor as well as the audience. We have similar experience with a musical performance. The value of a mathematical proof is not only in what it proves but also in how it is accomplished.

24. An artist is also a worshiper of consistency – visual and tensional consistency - of which visual symmetry is the primary expression. A round is a perfect symmetry. Other geometric symmetries that a beginner in arts have to master through vision only are given definitions in mathematics viz. a figure is symmetric if there is a rigid motion (translation, rotation, mirror reflection) which leaves the figure unchanged. Thus visual symmetry is obviously a meeting point of arts including architecture and mathematics.

25. But to an artist there are other symmetries too – symmetry in colour or in composition. At one point of creative phase the artist wants to free himself from the mathematical, perfect symmetry – then while on the left is a flower, on the right comes a bird, perhaps; a little more freedom places neither a flower nor a bird – there appear only some abstract forms. If the artist is more demanding, the human face breaks down, the two eyes no more remain in their natural locations. There is no end to this desire of an artist for liberating himself from the tentacles of traditional symmetry but at the same time he creates his own private symmetry, personal balance, which again he begins to dislike sometime in future.

28. Mathematics by and large, does not have any obligation to the ‘actual’ world. It creates its own objects to play with; it creates its own rules of the game. • “The part of mathematics so created does not (and need not) pretend to solve the physical problems from which it may have arisen; it must stand or fall on its own merit”, (Halmos). • This “merit”, apart from other demands, includes one necessary condition viz. the initial sentences, the axioms creating a particular object, must be consistent. Allegiance to consistency has so far been the grand narrative in this domain of human activity.

29. The notion of consistency however, is being challenged now-a-days and notions like partial consistency, consistency to a degree or localized inconsistency are gradually stepping into the arena. This phenomenon resembles to breaking down of classical, universally accepted symmetry or balance or consistency in arts and establishment of their very privatized counterparts.

30. Consistency is considered as the sole obligation of present day mathematics. Inconsistency, on the other hand, is absolutely not tolerated. Consistency and inconsistency are considered to be crisp notions – either ‘yes’ or ‘no’ type. In classical mathematics, there is no notion like partial or weak consistency.

31. I predict a time when there will be mathematical investigations of calculi containing contradictions, and people will actually be proud of having emancipated themselves from ‘consistency’. Ludwig Wittgenstein Philosophical Remarks (1975, p.323)

32. A mathematical artist may or may not be a user herself. She may remain sufficiently happy in the playful niche of mathematics away from market, advertisement, usefulness and applicability. And she may not be a seeker of truth nor desire for the Ultimate Meaning.

33. So, in her philosophy of mathematics, presence of opposite sentences may naturally exist. As a consequence, in the mathematical world that appears to the artist’s imagination, there is not only light but also shadows, there are not only strong and brave theorems but also theorems that are to some extent soft and shaky, those that have weaker provability; there are definite, infallible, symmetric constructions like the solar system and there are also abruptness like the sudden appearances of comets; as there are Cantor’s universes with the fragrance of Parijat flower as there are the incessant calls from deeply inconsistent black holes

34. And of course, a mathematical artist will put bounds around herself somewhere, but only relativistically and temporarily. These ties are consistencies created by her own self, they are her own responsibility – usually far removed and meta-mathematical – just as an artist delimits herself in a stability or consistency which is self-dug and impermanent.

35. The mathematics that we are thinking of would accommodate some amount of inconsistency or partial consistency in some places of its body. • Around these rebellious dens would persist disturbances, dissatisfactions and hence also a possibility of a Veena bursting into some musical resonance.

36. We have our cages too. These are our own constructs, testimonies of our own life – otherwise there remains only ‘existence’ – the existence of a stone – no live existence. But in our cages there do occur many holes, broken bars, loose ties, dark corners along with the lighted ones.

37. Thank You

38. Rupa • Rupa在《佛教大辞典》 吴汝钧 著（商务印书馆攻击有限公司 1992年7月台湾第一次印刷， 1995年9月北京第三次印刷）中有如下诸种解释： • 五因p111 • 以地、水、火、风四大种为能造之因，以诸色法为所造之果，此中有无义，称五因。道即是：（1）生因；生四大种所生之色，称为生因。（2）依因；色生起后，又随逐于大种，如弟子之依于师长，故名依因。（３）立因；任持四大种所造之色，如持壁画，名为立因。（４）持因；使所造之色，相续而不断绝，名为持因。（５）养因；增长四大种所造之色，名为养因。参见《俱舍论》卷７。 • 众生之生长有五义，称为五因。 • 五识身p128 • 身是集合，表示复数；五识身即5种认识机能的集合，这既是眼识、耳识、鼻识、舌识、身识的总称。Panca – vi == panaca kaya • 有离一切有为法，从理想一面说，都终能出离，而臻于涅磐的境地。即是说：一切有为法，都具有“涅磐性”。p220 • 色梵文Rupa。此词由意思为“形成形相”的动词rup所演变而成，解作“色是被造作形成的形相”。 Rupa是会变化，变坏的东西。故色专指那些具有形相、被生成的，变化的物质现象而言。传统即视之为变坏，质礙之意。另外，色又有颜色之意，这既是视觉器官的眼的对象，即五境之一，即为色界、色尘、色处。p220