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This paper first appeared in Philosophica Vol.74 (2004) pp. 3-29.
1 Wittgenstein (1953) uses a broader notion of choice in following a rule, claiming that any rule following involves an agreement or decision, in the sense that choosing to participate in a language game and form of life and maintain its rules and conventions, such as respecting Modus Ponens in mathematical proof, is always a matter of choice, not necessity. Without challenging this, the choices, conventions and decisions I am referring to are those that come about when there is no unequivocal and unambiguous already laid down rule to follow. Thus William Hamilton in inventing, constructing and defining the system of Quaternions chose to abnegate commutativity in the binary operation in his system (i.j = -j.i j.i) to obtain the best system with the properties he sought, although other options were open to him (Pickering 1995). This was not a permitted move in the contemporary language games of algebra, and thus led to the formation of new language games that have proved very fruitful in mathematics.
2 This is related to the need for premature closure noted and defined by Collis (1975). It represents the empirically noted desire of children in the early stages of learning mathematics to achieve syntactical simplicity by deriving a single answer in working a mathematical task.
3 Indeed Piaget (1952) had the prescience to identify the stage of being able to reverse informal mathematical and logical operations as a crucial step in the development of children’s mathematical thinking. The achievement of Invariance in this sense signals the transition to the stage of Concrete Operations.
4 My cautious phrasing is because in social theories of mind the personal need not be identified with the psychological. See, for example, Harré and Gillett (1994) Vygotsky (1978) and Wertsch (1997).
5 This is a key problem for versions of Intuitionism that claim that the objects of mathematics are personal concepts, but that different person making individual acts of construction arrive at identical concepts.
6 Phonetic language also points to the world of human vocal sounds, but this is irrelevant here.
7 I have to be careful what I assert here, because the diagrammatic and topological modes are representable in terms of the alphanumeric and the typological, just as pictures can be digitized. This follows from Descartes’ groundbreaking linking of algebra and geometry. However in making such translations and reductions a human faculty of knowing is eliminated. We have both spatio-visual and logico-linguistic modes of knowing (often identified with right and left brain hemisphere activities), and the elimination of one in favour of the other loses some of the balance, complementarity and power of our thought. The fact that all knowledge can be represented in binary code does not mean that human knowing is enhanced by actually representing all knowledge this way. Such reductions threaten or even destroy meaning and understandability.
8 The creations of mathematics are free in a strictly regulated and restricted sense that except in exceptional cases conserves existing meanings, rules and structures.
10 There are technical issues surrounding the relationship between truth and proof in mathematics foregrounded by Gödel’s (1931) seminal Incompleteness theorems that for simplicity I shall overlook here but that do not invalidate my argument.
11 It is also typically maintained by powerful social structures and institutions.
12 But note that for philosophical discussion purposes we might temporarily suspend belief in these or any assumptions, i.e., choose to play a different language game.