When a mathematician speaks about doing mathematics, living metaphors like ‘I move around in the mathematical waters’ (Atiyah) or the mathematical landscape, which Manin calls ‘mathscape’ become important.
The mathematician (in the mode described by Atiyah) behaves like a composer sometimes does. He plays around and when he finds something worth to be reported he reports it. But normally he does not report the rest! This rest plus the reported issues is called experience. Does the rest matter?
Yes, because it:
- provides tools for choice
- provides orientation for ‘moving around’
- leads to ‘feeling at home’, to familiarity
- provides connotations
- leads to intuition
Connotations are the basis for interpretations. Hence interpretation is only possible under the condition of experience. The same holds for evaluation. A mathematical issue can be evaluated only on the basis of experience. Everything else is in the domain of ‘checking’ and could most probably be automatized (up to the infinite regress of double-checking or checking the checker).
The main difference between a formal mathematical text or better a text which is freed from redundancies and presents a method or a result and a literary novel (or a poem or a drama) is that the latter (the novel) provides experience or better that through a novel experience is communicated. This is seldom done in mathematical writings. But this does not mean that it is impossible. Experience does not have to be linked to the ‘real’ world. It can be virtual (or fantastic) as well. My claim is that a novel, a narration, induces a similar feeling of experience like ‘real’ experience. If it allows, like a virtual reality game, to play around, to explore the (imaginary) domain, then the feeling of experience (and of presence) is even more real. It would be interesting to know if the brain can effectively distinguish between ‘real’ and ‘virtual’ experience.
The communication of experience necessarily includes the communication of emotions as well. A certain romantic element would enter. This romantic element might be desperately needed in order to overcome the increasing isolation of mathematics and of other sciences. Instruments are needed to share ones experiences. These instruments don’t need to be invented. They can be taken from the humanities and adapted appropriately.
In order to communicate the trips in the mathematical waters or areas coordinates, names, representations are needed. This holds not only for the communication of the trips but for the walking itself as well as for the looking around, the latter – moving around and looking around – being the ‘real’ actions. These actions need tools (or capacities/capabilities) in order to be performed. ‘Moving around’ needs also senses for orientation, for navigation and a ‘vehicle’ with a ‘motor’ or the ability to ‘walk’ (which means ‘technique’); ‘looking around’ needs senses for the recognition of ‘objects’ and for the evaluation of not only the objects encountered but also of the trips themselves.
There is an important complementarity hidden here. Moving and looking around needs senses but on the other hand senses are developed and refined by this moving and looking around. This seems to be a hermeneutical issue which means that other people are needed in order to start the process. These people have to initially show how to move and how to recognize something, how to communicate.
But there is a difference between communicating with somebody else and self-deliberance. Experience is something personal. This is why the arts have developed such a vast variety of tools in order to try to communicate experience. The achievements are big but since not all experience can be communicated (made explicit), the call to experience, to intuition, to habits has to remain only plausible (or probable).
Only I have the same experience as myself. But even this is not so sure because of the time factor in self-deliberation. The time-factor, the diachronic element, plays an important role in communication, especially in the communication through written texts/signs.
Since experience (and experience linked to mathematics in particular) is a process in time as well as a social process (at least its communication) we have an argument for the importance of two other disciplines from the humanities: history and sociology.
M. Atiyah source: An interview with Michael Atiyah, Math. Intelligencer 6 (1984), 9-19):
[…] I just move around in the mathematical waters, thinking about things, being curious, interested, talking to people, stirring up ideas; things emerge and I follow them up. Or I see something which connects up with something else I know about, and I try to put them together and things develop. I have practically never started off with any idea of what I’m going to be doing or where it’s going to. I’m interested in mathematics; I talk, I learn, I discuss and then interesting questions simply emerge. I have never started off with a particular goal, except the goal of understanding mathematics.
Y.I. Manin: (source: ‘Contribution in panel discussion on “The theory and practice of proof”, 1, Proceedings of the seventh International Congress on Mathematical Education 1992, Montreal, Canada):
My mental eye sees something like a landscape; let me call it ‘mathscape’. I can place myself at various vantage points and change the scale of my vision; when I start looking into a new domain. I first try a bird’s eye view, then strive to see more details with better clarity. I try to adjust my perception to guess at a grand design in the chaos of small details and afterwards plunge again into lovely tiny chaotic bits and pieces.
Any written text is a description of a part of the mathscape, blurred by the combined imperfections of vision and expression.