Differences induce communication. Communication seems to be reducing differences (what about entropy?). On the other hand communication needs a common code (for reading) as well as common experience (for understanding or interpreting/translating). Communication has to start and it has to be evaluated: Did it actually take place? (question-answer-mode) Was the message understood? (How does one check this?) Did the communication lead to a reduction of entropy ‘between’ sender and receiver? How do we check this (make this sure) in the case of mathematics? How does the size of the difference between sender and receiver affect the communication? What is it that makes the difference? Is it training, language/code, experience? The notions seem to be very near to each other – especially experience and communication. The order of magnitude seems to make the difference. How can experience be communicated or let’s be more specific: How can previous communication be communicated? It is all about the communication of communication and only the roles of sender and receiver differ. It is a quest for transitivity but maybe also for alteration and amplification. There seems to be a difference between communication in the domain of the arts and in the domain of the sciences but the location of mathematics with respect to these two domains is unclear. V. I. Arnold claims that mathematics is a part of physics (the cheapest part). But on the other hand it is all about relations and takes place in one’s head only, which makes it more a part of the arts where one seeks to communicate about previous communications between humans.
The most evident difference is the difference in experience. This is on one hand a difference between two humans but also a difference between one and the same human being in time. I am different from what I was some time ago so it is reasonable that I would like to communicate with myself – by writing something down or by making some kind of record. This communication is one-sided since only the past/present can serve as sender and only the present/future can serve as receiver. This one-sidedness is partly resolved by dialogue, by reducing the time-interval to a minimum. The dialogue can be between two or more individuals (dialogue or discussion or performance) or ‘between’ one and the same individual (‘talking’ to oneself). In the latter case there is a certain problem concerning the required difference. But the question is whether the latter case has to do something with what we call ‘thinking’? Rotman speaks about ‘thinking’ and ‘scribbling’ when he defines mathematics. Is the difference between these two activities: thinking and memorizing (externally through scribbling or internally (through what?)) only an issue of time? But maybe there is a qualitative difference when the time-interval in communication gets under a certain threshold. The issue of the ‘present moment’ and of presence in general seems to arise quite naturally. The main role of memory in the context of communication is that it enables decoding.
The above text is quite general. A consequence concerning mathematics is that it is much more important to communicate ‘living mathematics’ instead of ‘dead mathematics’. This means that messages should be not about results or about formal definitions but about how to approach them, how to arrive at certain ideas, how to acquire and practice a certain technique, how to move in the landscape (‘mathscape’ in the words of Manin) in order to make the same or similar experience as the sender of the message. It is exactly the communication of previous communications of the sender – communications with other mathematicians, communications with oneself (thoughts, ideas, mistakes, etc.) and communications with ‘dead mathematics’ (interpretations, grouping and structurizing, rigorizations or its opposite – making notions or techniques more flexible, more vague). The last item – communication with ‘dead mathematics’, is still unclear for me. I think that it makes no sense. It is part of the first item, of communication with other mathematicians, not orally but through written texts which have undergone some process of external evaluation. Here we arrive at an interesting point, namely that much too often the written texts are not written primarily for other mathematicians but for the evaluators (who, of course, are also mathematicians). This problem is discussed in the entries on critique and judgment.
An additional important aspect (more linked to art than to mathematics, but I don’t know) is that there seems to be a difference between a record (video, audio, text, memory) and the ‘real time’ experience. The question concerning mathematics would be: What is the difference between face-to-face (or real-time) communication and communication by recording devices (media)? Is there a difference in mathematics going beyond the elementary aspect of being able to ask questions immediately? Is there something like ‘atmosphere’ like ‘emanation’? Why do people learn quicker through direct communication?