The notion of the concrete is a relative notion. It is linked to the notion of the abstract but it seems that in our context of ‘living mathematics’ the concrete might have higher priority than the abstract. What do I mean by this? I mean that concreteness is linked to experience. Experience makes abstract notions more familiar and hence more concrete. Only experience enables a mathematician, after having become familiar with an initially unfamiliar domain of notions and their interrelations, to think and perform a next step of abstraction. This iterative process is described in the entry on abstraction.
It seems that ‘concrete’ and ‘abstract’ are not complementary, they are not opposite notions. Abstraction is a more mechanical process, maybe it could be automatized. This is absolutely not the case with ‘concretisation’ through experience, because it is absolutely unclear how experience might be automatized. The metaphor of a ‘process’ seems unsuitable for the notion of ‘experience’. The transition from the unfamiliar to the familiar, from the ‘non-concrete’ (I start more and more to dislike the use of ‘abstract’ in this context) to the concrete cannot be linked to a process.
Concreteness is strongly linked to the feeling of presence. In this context it is interesting to note that metaphors seem to play a crucial role in the above mentioned transition from abstract or better from ‘non-concrete’ to concrete. It is exactly metaphor and not analogy or similarity that evokes the feeling of concreteness, at least in the initial steps of acquiring familiarity but much more when this kind of concreteness, of experience is to be communicated. Something that is abstract or non-concrete or ‘not experienced’ to the receiver and concrete or present to the sender has to be transmitted with a metaphor because the metaphor links the unfamiliar, the non-concrete with something that has a certain (at least assumed) status of concreteness for the receiver. Now I have difficulties to find suitable examples from mathematics. There is a whole entry about metaphors in this encyclopaedia but the good examples are missing. The article ‘Mathematics as Metaphor’ by Yu. I. Manin does not help in this situation since it treats the ‘whole’ of mathematics as a metaphor. This is quite interesting but in our case we need examples of specific metaphors used in order to transmit mathematical experience.
How is the issue of concreteness dealt with? In contemporary expository mathematical texts there is something called ‘examples’. These parts of the text serve to make the introduced notions more concrete. It is not clear if the examples are a part of ‘dead mathematics’. They rarely appear in research papers. Does this mean that research papers are directed to people (to receivers) for whom the notions used have already a certain level of concreteness?
It would be extremely interesting to compare blog discussions on a mathematical topic or the discussions in the development of a polymath project, a collaborative research project, with its final outcome – with the published paper which has resulted out of the project. This comparison should be made with respect to concreteness. It would also be interesting to compare the initial discussion and the final paper with a subsequently written expository text on the same topic. Can the initial discussion serve as an expository text or not?
Another aspect of concreteness would be to compare, in the context of familiarity with a mathematical item or with its recognisability, the issues of checking and recognition. Can we speak of something like a transition from the checking of individual properties given e.g. in a definition of a mathematical item to its instant recognition in the sense of gestalt theory? I am not sure if this last issue fits here or if it should be placed in another entry of the encyclopaedia.
Another, much more general aspect of concreteness (and this seems like a strange paradox) is the simple issue that the work of a specific mathematician on any kind of mathematical problem is something very concrete. Living mathematics itself, the thinking, the interaction with the dead text, the decisions to go in this or that direction, the lightbulb phenomenon or the ‘aha’-phenomenon (‘invention’, if you like) are concrete events in the sense of the event of playing and/or listening to a piece of music but also in the sense of speaking and/or listening at a very specific moment. Rational description/discourse is not designed (suited) to deal with specific acts.
Another possibility to speak about the above issue is by using the terms ‘individual’ and ‘general’ (or ‘interchangeable’ or ‘non-distinguishable’ or ‘equivalent’ or ‘ideal’). Mathematics seems to be an activity, a language, which serves to ‘describe’ the non-individual and all its similar notions. Mathematical items are most often equivalence classes which consist of interchangeable sub-items. A simple example is a vector but a number seems to fulfil the requirements as well. A ‘point’ in space is on one hand timeless and hence has no history and on the other hand it is not distinguishable from any other point exactly in ‘his’ ‘pointness’.
A similar issue arises in physics but with much more dramatic consequences. The issue of the loss of individuality of a physical object (like an elementary particle) is much more problematic because in mathematics we can avoid speaking about objects (and speak about ‘items’ instead) which does not make sense in physics.
Romantic mathematics is much easier to imagine (because of its similarity to art) than romantic physics.