Explanation

What does ‘self-explanatory’ mean? Is there something like a ‘self-explanatory’ exposition or (re-)presentation? Is ‘self-explanatory’ the same or similar as ‘self-evident’?

Explanation to oneself and/or to someone else is a part of the communication process viewed from the sender’s point of view which has the intention of achieving understanding on the side of the receiver. Explanation depends on both parties and hence cannot be formalized. It does not make sense to speak e.g. of an ‘objective’ ‘explanatory power’ of a proof. Explanation is more suited to be considered an art.

Like in every domain of the arts one can try to criticize specific cases of explanations or trends/styles in explanation. The issue of taste plays a necessary important role. Some aspects of an explanation might be more objective than others and one can speak of a mathematical item being explained well or badly. One and the same explanation might be good (explanatory) for one person and poor/bad for another.

Explanation is clearly a part of ‘living mathematics’. But isn’t it also something like a genre in writing, in literature? Is there something like ‘explanatory literature’?

Mathematicians are adapted, used, trained to understand specific modes of explanation. They receive this training from specific persons (teachers, professors, collaborators). This is part of the process of initiation into the mathematical community. Gradually one starts to explain in a similar manner as one’s teachers/colleagues. On the other hand explanation needs feedback from the receiver about its success which leads (or may lead) to a successive adaptation towards the specific audience (of receivers).

Is understanding more than just an exact repetition of the explanation? Isn’t understanding linked to the ability to modify the explanation, to produce an alternative explanation? A successful explanation would then mean the production of alternative explanations. The effort of the receiver has to be to produce a new explanation so that the sender says OK!

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Experience

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.

 

Testimonials:

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.

Hermeneutics

If we are interested in the communication of mathematics (or better: of mathematical understanding) and think of a very simple model consisting of a sender, a message and a receiver then hermeneutics is dealing with the part of the receiver. The massage might be intended or not, the important thing is that it is considered by the receiver as a message. This message has to be interpreted (which means: translated) which induces a difference (big or small) between the code of the message and the codes at the disposal of the receiver. It is even not clear if the code ascribed to the message reveals the actual code of the sender. Mathematics (and especially the part of it called logic) is the science which tries to avoid this problem of interpretation but since humans are involved it cannot be successful. To show this is the main aim of this encyclopedia.

Critique

How can a piece of mathematics be criticized? Concerning its correctness there cannot be anything more than a YES or NO.

Who are the critics? What is the status of self-criticism? Does self-criticism lead to a self-reference or to an infinite regress? Who criticizes the critics? The issue of critique leads to the issue of authority as well as to issues like anonymous vs. non-anonymous and private vs. public critique. ‘Objective’ critique is another idealistic phrase: can critique become automatized and will automatized critique be more than just a local checking of the correctness of a proof?

The (by now) hidden process of refereeing seems to be crucial for detecting specific characteristics of ‘living mathematics’!

What is the difference between an ‘authorized’ referee and an ‘ordinary’ reader? This is a main question!!

If the often proclaimed unity of mathematics was the case then there would be also a certain unity in the criteria upon which a piece of mathematics is criticized. But the problem is that there are actually many sub domains with their own more or less separated communities and thus with their own values. The separation is best seen in what is called school-mathematics versus university-mathematics. There might be a similar but much smaller separation between undergraduate mathematics for non-mathematics students (engineering but also business studies, biology, medicine, etc.) and mathematics for future mathematicians.

This seems to be the result of a wrong belief – mostly met in mathematics and in the sciences as well – that correct statements ‘speak for themselves’. This is an important negative consequence of the split between the sciences and the humanities. Critique serves not just for quality control and for evaluation. It serves also to create a channel of communication between the author (the sender) and the audience (the receiver). It is a message about a message, so there is an issue of self-reference. Not only because of this but also because of the mediating function of critique it is clear that it does not make sense in a closed system. It actually serves to ‘open’ the system. The ‘insiders’ don’t need critique. They agree because they share very similar values and experiences. They only nod and the type of nodding depends upon whether the (mathematical) item in question has been appreciated or not.

It would be interesting to look at cases where the members of a specialized mathematical sub-community disagree. I would suppose that the disagreement would be more about the correctness than about the quality. Here we get back again to the reviewing process. Disagreement in mathematics is not allowed to become public. It is either ruled out by not accepting a specific paper for publication or the issue of disagreement gives birth to a new subfield like nonstandard analysis or like constructive mathematics. Maybe nowadays through the many platforms for informal discussion the issue of disagreement could be studied on a more concrete basis.

Philosophy

Mathematics as well as philosophy have a very long tradition which results in several problems hindering the communication between each other. Both have developed many subfields some of which require long periods of study in order to enable a person to both understand some of the main problems in both fields as well as to acquire the tools with which to approach these problems.

This is important in the sense that the respective scientific sub-community bases its values on the fluency in its respective language including the familiarity with its sophisticated objects and tools.

This leads to three problems:

Firstly if a philosopher starts to deal with mathematics, she is most often not taken seriously by mathematicians because of the lack of fluency in the mathematicians’ field or subfield. “You don’t understand what I am doing so you cannot tell me anything of relevance for my work.” We should not neglect this argument but take it seriously, independently of the quality of the philosophers work!

The same holds the other way around: if a mathematician starts to do philosophy (this holds for history as well and much less in education which will be treated later on) the general impression of philosophers is one of dilettantism which also has to be taken seriously.

This should remind us that neither philosophy nor mathematics is something easy to grasp and learn and that it takes time to be enculturated into the respective communities and hence to be taken seriously by its members – to make them listen and adhere to ones arguments.

Secondly both philosophy and mathematics are quite segregated in themselves. Both have a big variety of sub-disciplines and a simple combinatorial question arises: Can we do philosophy of mathematics in general or do we have to do analytical philosophy of the axiomatic method or phenomenology of geometry or hermeneutics of topology or any other combination of sub-disciplines in both mathematics and philosophy?

This raises the immediate question of the unity of both disciplines. What is philosophy about (or better: what should it be about) and what is mathematics about? Only if we can find those unifying items we can attempt to bring them together. Of course this makes only sense if we have solved the first problem. Or maybe it could even help us to solve the first problem!?

Thirdly these ‘communities of practice’, however scattered they might be, have developed historically a set of questions, problems, methods, theories and programs (and a system of values!). These are constitutive for the respective community and hence of enormous value – hence everything outside of these enterprises needs a big political effort in order to be introduced and considered as a valuable question, problem, method, theory or program.

The approach to this last problem, though difficult in itself, seems clearer than the resolution of the other two. This problem is also not as symmetrical as the first two ones. It deals more with the question how to enter (as a philosopher) into the mathematical community or into a certain mathematical sub-community.

There is another issue which is more intrinsic to both domains. There are difficulties in accepting each other due to the different nature of approaching problems. Mathematicians search for solutions. Philosophers very seldom provide solutions to the problems they pose. If they happen to provide solutions, they are not of the type mathematicians are used to. Another and maybe less important aspect is that mathematicians are used to formalisms and the level of formalization is also a value in itself. Very little of philosophy is formalized and it is by no means a goal there.

Mathematicians are used to use diagrams and sketches and to make further and further abstractions in order to generalize their statements more and more. Maybe an attempt to use a more diverse set of representational tools than only ordinary plain prose could be a chance to make philosophical texts more appealing to mathematicians.

This difference in the approach to problems is on one side a big problem, on the other side it leads sometimes even to animosity – at least from the side of mathematicians. Statements by prominent mathematicians are quite strong in this respect. Here are two examples:

The following remark is attributed to Karl Friedrich Gauss: “When a philosopher says something that is true then it is trivial. When he says something that is not trivial then it is false.”

Contemporary mathematician and Fields medalist Jean Bourgain used during a lecture (‘Visions in Mathematics’, Tel Aviv 1999) the phrase “this is a philosophical remark” for everything which was not directly related to the mathematical demonstration. The strange thing is that he could have avoided (could he really?) all these unnecessary commentaries but he didn’t do so. Nevertheless the impression was one of a very dismissive attitude towards those “philosophical remarks”.

The good news is that the existence of animosity (and not just ignorance) indicates that there might be more points of contact than initially supposed by both sides.

  • A certain number of questions arise out of the above thoughts:
  • Are there aspects in mathematics which need a philosophical treatment? Where the issue is not (only) to find a solution?
  • Are mathematicians aware of these problems?
  • How can the situation be changed so that these problems start to be perceived as relevant?
  • How can these problems become important?

Obviously a political effort is needed. An effort has to be undertaken aiming at a change of values or of the hierarchy of values of the mathematical community (or of a mathematical sub-community). But for this task we should know the current values of the mathematical community, their hierarchy and their changes in the past.

Can philosophy also profit from this? Who is the ‘queen of sciences’? In this respect philosophy has been successfully contested by mathematics! The consequences from this are well presented in the two-cultures-debate started by C. P. Snow.

There is another important problem. Philosophy is not anymore the sole science/domain which might be useful to reflect upon mathematics. It has split – and this has various extrinsic reasons – into Philosophy, Social Sciences like Sociology or Anthropology, Psychology, Cultural Studies, History, Semiotics, etc. It is really a mess! And this mess is also reflected in what has been published as reflections upon mathematics. There are sociological studies (Bettina Heintz), psychological attempts (Hadamard, Borovik), attempts from the standpoint of semiotics (Rotman, Marcus, myself), history of mathematics develops quite well in recent years (M. Epple, L. Corry and many others), etc. All this scattering seems to be a problem of philosophy as well.

Does the publishing of more and more texts mean that we are on a good track? The pity is that mathematicians still don’t care.

In order to approach these questions I first must clarify what I mean by philosophy and what I mean by mathematics.

Philosophy has initially been an enterprise to reflect upon the world we live in. It has developed a big number of tools which provide a language but also techniques to describe, evaluate, make comparisons, etc. One big advance in philosophy during the 20th century was that issues of communication and of language have been given a much more prominent role than before (Semiotics, The New Rhetoric and Wittgenstein’s work) and that aspects of training and habituation have become an issue in epistemology (Polanyi’s work). It is interesting that both aspects are linked to time which seems to be in conflict with the claim of universal and hence time-independent insight.

The quest for universal insight is a main driving force in both domains – philosophy and mathematics. But an insight which is detached from anybody who has eyes to look and see does not make much sense. We could imagine a trivial thought experiment: does a library with all the universal insights but without any readers make sense? There will be nobody around to judge or even to approve, that there are really universal insights available in this library.

From the very moment when a person, a reader, a partner in communication enters the scene all issues of communication become indispensable.

It turns out that it would be more useful to consider all these universal insights as tools. A tool is something that becomes relevant (as a tool) from the moment somebody begins to use it. Before this moment it might be something that could be observed but nothing more.

In order to start to use the tool one needs to know how to use it. It needs some kind of instruction. This means that somebody has to give (write) an instruction and somebody else has to learn from this instruction and to practice. The latter includes of course all aspects of learning and training.

We may conclude that universal insights in the domains of philosophy and mathematics seem to be universal only for the learned and trained ones.

It is important to know that I do not want to enter a debate concerning the objectivity or subjectivity of results in both domains, especially in mathematics. This is the reason why I use the term “insight” – a notion which is clearly human-related. By the way, it would be interesting to ask the question, whether a computer or a computer program might have something like an insight and what this might mean.

The discussion of insights gives me the opportunity to introduce a tool which comes into use in some of the items discussed in this encyclopedia. The tool I mean is the use of self-referencing as a means for reflection.

In our case the issue is to gain insight about insights. It may sound strange at first but it works. This is a type of self-reference which does not lead to an infinite regress. If we have gained insight about insights we shall call this a ‘second order insight’. An insight about a second order insight is itself again a second order insight. The point is that there is a difference to the kind of self-reference which leads to infinite regress: a second order insight leaves the potentially closed system related to the first order insight. The tools used in order to gain a second order insight can be used again for gaining an insight about this second order insight. The issue is that when a second order insight is investigated it is itself treated as a first order insight.

To describe the above situation in plain words:

  • Step 1: we have understood a mathematical relation.
  • Step 2: Next we want to understand what it means to understand a mathematical relation – we want to understand what understanding means.
  • Step 3: After having achieved this, the question of understanding what the understanding of understanding means does not bring any new understanding different from the one in step 2.

The method of self-reference shows in general, that a closed system is not possible. If self-reference leads to paradox, it means that a system has been artificially considered to be a closed one.

There is a fine and profound example from biology: A caricature shows a scientist who rushes into the room to tell his colleagues that he has discovered the gene which determines the belief that everything is determined by genes.

Mistakes

What role do mistakes play in the everyday work of a mathematician?

Types of mistakes:

  • Logical mistakes, wrong calculations – easy to check (?!)
  • Wrong tactics – choosing manipulations which don’t deliver intermediate solutions
  • Wrong strategies – although intermediate steps might work, the chosen path doesn’t lead to the intended result

Good and bad mistakes:

  • Are there good and bad mistakes?
  • Obviously there are good mistakes, as remarks by G. Shimura about Y. Taniyama and remarks by G. H. Hardy about S. Ramanujan suggest.
  • How can a good mistake be recognized?
  • Good mathematics does not necessarily need to be correct mathematics. Obviously good mistakes are good mathematics.

Communication of mistakes:

  • The work on a mathematical problem contains mistakes.
  • This holds for every mathematical problem and for every mathematician.
  • The proficiency in coping with mistakes constitutes a crucial part of the expertise of a mathematician.
  • Mistakes are banned from published papers
  • Mistakes have no place in dead mathematics
  • Mistakes are seldom reported, especially in modern mathematical practice
  • There is maybe a change of paradigm on the way, related to the polymath projects.
  • standards should be developed for the form of a paper which presents unsuccessful work but also the rhetorical and stylistic tools will have to be different from those for “correct” papers
  • standards for the evaluation of such papers would be needed (how should such a paper be reviewed)
  • the rhetorical (and stylistic) tools will have to be different from those used to present successful research

How can the communication of a strategic mistake look like? “I tried to relate field A to field B. Don’t try it again, because of C.”

  • C should contain important wrong mathematics
  • The communication of C can result in
    • Others finding successful paths
    • Other but different wrong attempts
    • Questioning of some intuitions
    • “declaring” this relation as non-existing (or at least as non-productive)

Concerning my own interest in a problem from number theory, an expert (E. Hlawka) told me not to work on it: “Don’t burn your fingers with this!”. Obviously he had some reasons for this which he did not or could not make explicit.

What are the reasons for a certain problem to be considered as “hard”? Is anything like this communicated formally, thus published? Can the notion of “hardness” be formalized?

Maybe from refereeing each other’s paper submissions which were eventually rejected. But these are only those attempts where the author could not find the mistake himself. I would suspect that a mathematician will find most of his mistakes by himself (compare to chess) and will not submit them for publishing.

Can a mathematician judge the quality of his own mistakes? “I made this mistake but it was a good mistake and so I will write a paper about it.” “This other mistake was a “bad” one. Forget it!”

Are there “typical” mistakes in research mathematics like there are in school mathematics?

Further isues:

  • Mistakes and understanding
    Wrong vs. correct understanding
  • Inconsistent conventions (important??)

 

Testimonials:

Tanyiama was not a very careful person as a mathematician. He made a lot of mistakes. But he made mistakes in a good direction and so eventually he got right answers. I tried to imitate him but I found out that it is very difficult to make good mistakes.

I have in the past tried to say things like “his failure was more wonderful than any of his triumphs”, but that is absurd. It is no use trying to pretend that failure is something else. All we can say is that his failures give us additional, surprising evidence of his imagination and versatility. And we can respect him as one who let his mind run free, instead of keeping it under saddle and blinkers like so many others do.

Mistakes are an important and instructive part of mathematics, perhaps as important a part as the proofs. [… in the text Arnold gives examples of mistakes by Leibniz, Cauchy, Lagrange, Poincaré, Hilbert, Burnside, Petrovsky, Komogorov, Pontrjagin and Rokhlin]

Memory

Human memory plays a constitutive role in at least three mathematical activities:

  • During the following of an argument, of a proof, one has to remember if not every so at least a certain number of steps in order to be able to see that a connection between certain mathematical items is established by the proof.
  • More interesting is the role of memory in making analogies or comparisons between mathematical items which can lead to the statement of questions, problems, conjectures or hypotheses.
  • The previous point can be expanded to the activity of finding arguments in order to answer a question, to solve a problem. If the argument is new, maybe there is no memory needed but in most cases an already known argument is borrowed (because it seems to fit the structure of the problem) and maybe modified (or the problem is modified in order to fit the argument). Nevertheless the argument (or technique) has to be retrieved from ones memory.

A last point concerns the relation between memory and habit, between memory and know-how. Maybe it is similar to the relation between memory and spoken language. It seems that the main issue is the time of retrieval of ‘information’ and probably ‘information’ is the wrong word, but anyhow it seems that this retrieval-time is minimized to something we call an ‘instantaneous reaction’. It would be interesting to project this to computers – it seems not to work, since there is always a ‘process’ of information retrieval from the memory (of the computer) which seems not to exist in what we consider to be an instantaneous reaction, something like ‘real-time-thinking’ (this last expression is maybe a pleonasm).

Items

Abstractions in general and mathematical abstractions in particular are much better described as ‘items’ than as ‘objects’. The advantage is that it is on one hand much nearer to how abstractions are treated by mathematicians and on the other hand no issue of existence arises. The whole issue of Platonism arises from the treatment of abstractions as objects without having any reason to do so. This leads immediately to the Tausendfüßler-Paradox: mathematicians create their own world exactly in the same way in which a child creates an own world when listening to a fairy tale. For the child there is no difference between fiction and reality. The fictional story is perceived as ‘real’ and there is nothing wrong with this. Every request to speak about the story, to explain the story will pull the child (the mathematician) out of the fiction. When the child matures, it is less and less able to ‘dive’ into the fictitious world of a fairy tale. Of mathematicians (and, by the way, also of musicians) another development is expected – they should retain as much as possible their (childish) ability to dive into a fictitious world. A multitude of such ‘worlds’ can be created from (mathematical) abstractions by interpretations.

The ‘characters’ of such a story of fiction, which might be called a ‘piece of mathematics’, are ‘identifiable items’ and not ‘existing objects’. By the way, it is very difficult to avoid becoming immediately linked to ‘social constructivism’ with such a line of thoughts although it is not so clear how this ‘diving’ into the story by a child or by a group of children should be classified.

How does this relate to the physical world? What about Wigner’s ‘unreasonable effectiveness’ of mathematics in physics? Maybe it is in the process of abstraction. Mathematical items are derived from properties of a physical object or of a conglomerate of physical objects. Why should such a fairy tale, located in the fictitious world of such abstractions not lead to something that can be translated back as a hypothetical property of a physical object? Maybe this is the way the whole enterprise of science works? But then the effectiveness of mathematics is reasonable as much as abstraction is reasonable and as a consequence – as much as every kind of representation including language is reasonable.

Forgetting

A proof has been understood when one can forget the details and still be able to reconstruct it, if needed. Only some ‘supporting points’ remain to be remembered (the German word ‘Stützpunkt’ fits very well here).

Another aspect of forgetting is linked to habits and experience. After a year of continuous training a swimmer cannot remember how she approached breast stroke one year ago. Even if she watches a video of herself, this will be another person swimming. She will be able to remember a few things but she will not be able to remember her approach to breast stroke from that time. The same holds for playing a musical instrument. And the same holds for doing mathematics. A person seems to have only one attitude only one approach to a certain domain at a certain time (even if this approach consists of a variety of strategies). The point is that through dedicated training the approach changes and the old one disappears – I cannot think like I did one year ago. Of course by ‘one year’ I mean a reasonable time-interval during which new habits are acquired – this depends strongly on the person.

Testimonials

W.T. Gowers (source: http://gowers.wordpress.com/2009/01/27/is-massively-collaborative-mathematics-possible/):

A small remark about the previous comment: there are numerous examples of singly authored papers where the author (after a suitable time lag) doesn’t really understand the paper. There’s even a famous example (the details of which I’m afraid I’ve forgotten) of a result that was independently proved by three people of whom two were different time slices of the same person.

Concentration

The amount of concentration, of time spent (by mathematicians) in concentration on mathematical issues is very big. The unconscious effects need the previous concentration time.

Concentration vs. communication: is there any influence to communication during concentration? Is there a difference between being alone concentrated on something on one hand and discussing this thing with others (or presenting/performing it for others) or the concentrated perception of a lecture or talk? There is an obvious analogy to sports and performing arts or to the active part of non-performing arts.

Concentration is linked to attraction (positively) and to distraction (negatively). It is linked also to ‘absorption’ – ‘to be absorbed with something’ as well as ‘to be focussed’ on something, to focus one’s mind.

Is there something that might be called the transcendence of concentration? Sometimes we see this in performing arts when the artist becomes ‘one’ with the performance. There is no special notion for this ‘unity’. It seems similar to the proficiency in using a language – one stops concentrating on the language but uses it ‘fluently’ in order to express something with it. And moreover – not just to express something with it but also to identify oneself with it. Can this be the case in mathematics as well, if we consider it a form of art, or at least having artistic aspects? Is this linked to the unconscious, to insight? Is art the expression of insight? There are very few testimonials about this state of ‘unity’. The pianist Friedrich Gulda has described such moments by saying that the feeling is somehow that he is no more playing by himself but ‘it’ is playing. It would be extremely interesting to link such experiences made by artists with similar experiences made by mathematicians. By the way, it does not mean that in such moments the artist does not control the situation. It is something like a controlled state/flow of trance – so concentration is still there. There are amazing (but very few) depictions of such situations on video during music performances e.g. by Glenn Gould, Thelonious Monk or Wilhelm Furtwängler.

What is the emotional aspect of concentration? Is it linked to attraction, to passion, to devotion, to dedication? Are these the true criteria for good vs. bad mathematicians (hence mathematics)? Curiosity might also play a role but it seems secondary to me compared to the just mentioned criteria.

This item is linked to training and to commitment.