Epilogue
Ludwig Wittgenstein wrote about the fundamental mathematical act of inferring*:
When we ask what inferring consists in, we hear it said e.g.: 'If I have recognised the truth of the propositions..., then I am justified to further write down...' — In what sense justified? Had I no right to write that down before? —' Those propositions convince me of the truth of this proposition'. But of course that is not what is in question either. —' The mind carries out the special activity of logical inference according to these laws.' That is certainly interesting and important; but then, it true? Does the mind always infer according to these laws? And what does the special activity of inferring consist in? — This is why it is necessary to look and see how we carry out inferences in the practice of language; what kind of procedure in the language-game inferring is [my emphasis]
(Wittgenstein 1978 p.43)
This work is an exploration into these 'kinds of procedure' with regard to the novice mathematician. It started as generally as indicated by this quotation and in the process it was refined into the spectrum of themes synthesised in Chapter 10. Further refinement and substantiation of these themes is necessary.
I take inferring in the above to mean more than simply deduce via the rules of logic. I see it as both reasoning formally and meaning-making. In this study these two acts were seen as the components of starting to act mathematically at an abstract level and they were shown problematic.
Mathematical abstraction 'supplies us with a new picture, a new form of expression' and 'there is nothing so absurd as to try and describe this new schema, this new kind of scaffolding, by means of the old expression' (p.138). Not because 'the finite cannot grasp the infinite' (p.263) — what is the point of attempting to understand then? — but because a new language is introduced in order to express new meanings. In advanced mathematics a large number of these new meanings are about the multiplicity of representations and the effectiveness of flexibly alternating between these representations. The novice does not always seem to attribute this type of significance to the new language and tries to acquire the new meanings by trying to describe it in terms of the old pictures and the old words. All the new language is about is translating 'vague ordinary prose' into clear — hence usable — expressions. The novice is, to start with, an applied mathematician: nothing makes sense unless it has a purpose. Concept formation then is the search for 'the limit of the empirical' (p.237) and, this, didactically, possibly implies the necessity to treat learning as an empirical extension.
The above are, simultaneously, findings of this study and challenging questions. I note that if 'we ask these questions at all, this points to the fact that the answers are not ready to hand' (p.133) and because 'philosophical dissatisfaction disappears by our seeing more' (p.118) I can only wish for more as the way ahead.
*
Wittgenstein, with his particular interest in mathematics and its psychology, has been a constant source of inspiration in this study. All the references in the Epilogue are his words from (Wittgenstein, 1978)