Rather than presenting absolutist and relativist philosophies of mathematics as logically inconsistent, I suggest that each represents an aspect of mathematics when viewed from a particular perspective. Davis and Hersh (1981) use the analogy of a cube looked at from different angles. They argue that
[M]athematics is not one single thing. The Platonist, formalist and constructivist15 views of it are believed because each corresponds to a certain view of it, a view from a certain angle, or an examination with a particular instrument of observation…Since they represent views of the same thing they are compatible (pp. 396, 397).
As in the introductory quote from Latour’s (2013) analysis of the nature and development of knowledge in science, I suggest that looked at from far away
15 Constructivist in the sense of the mathematical philosophy of constructivism as described, for example, by Brouwer (Bridges & Mines, 1984), in which the only mathematical objects that are said to exist are those that can be constructed by finite means. This is not to be confused with constructivism as a theory of learning or with social constructivism as a philosophy of mathematics education.
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mathematics appears outside, unified, inanimate and undisputable. Yet looked at from close at hand in the act of doing mathematics it is internal, multiple, animate and disputable. Absolutist and relativist philosophies should thus not be seen as mutually exclusive, but rather as essentially differences in representational grainsize (after Manders, n.d. cited in J. R. Brown, 2008), or outside and inside views of mathematics respectively. Just as diagrams, text or equations can each represent the same mathematical object, absolutist and relativist philosophies refer to the same activity and discipline. Just as diagrams, text or equations each reveal or obscure particular aspects of mathematical objects, so absolutist and relativist philosophies reveal or obscure certain aspects of mathematics. Absolutist philosophies, like diagrams, have larger grainsize in that they are highly effective in showing the big picture and part-whole relationships. They are mathematics seen from the outside. However, they obscure the fine details revealed in the development of mathematics, just as diagrams can obscure details such as exact lengths of line segments (J. R. Brown, 2008). On the other hand relativist philosophies reveal the inherent uncertainties in the development of mathematics, but obscure the big picture of the logical interweaving of mathematical concepts. They are mathematics seen from the inside.
The differences in points of view are vividly revealed in Burton’s (1998) interviews with 70 practising research mathematicians in the UK. She found that it was “impossible...to speak about mathematics as if it is one thing, mathematical practices as if they are uniform and mathematicians as if they are discrete from both of these” (p. 141). Many of themathematicians in her study used metaphors of geography or jigsaw pieces to refer to the task of locating their work within a big picture. For some this provided a personal dilemma, as they were caught between wanting to subscribe to this big picture image of mathematics as a discipline and the socially constructed research activity in which they were engaged.
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Mathematicians do mostly subscribe to an absolute view of mathematics and what I have called The Big Picture lies not far away from where they position themselves. On the other hand, they speak in very different voices about the nature of the enterprise upon which they are engaged making clear how integral to it is both their own person-ness and the nature of their professional interactions (Burton, 1998, p. 140).
The big picture, or as I have termed it the external view, dominated these mathematicians’ approach to teaching, as I suggest it does for most teachers of mathematics in schools. Few would argue that obtaining a big picture of mathematics is unimportant—indeed, without a big picture it would seem that there is little point in seeking mathematical knowledge through social activity. However, Burton found that the fine-grained or internal view of mathematics as a human activity was “entirely lost in the ‘objective’ mathematics they, as teachers, thrust towards reluctant learners” (p. 140).
The inside and outside views of mathematics are very much in evidence in the Narrative of Slowness introducing Part 1. By exploring numbers that could or could not be represented as trapezia I was able to make and test some tentative hypotheses. What was interesting about those hypotheses was their connection to other mathematical ideas, a connection that was not immediately obvious from the context of the problem. I wanted to understand these connections, so set about proving the result. This was the human endeavour of doing mathematics. However, at all times I was conscious of the big picture of mathematics. If I had not expected the pieces to fit together I would not have sought a solution to the problem, nor a proof of the result. The personal satisfaction gained from the problem was thus a result of both internal and external views of mathematics: without engaging in the act of mathematical exploration, discovery and explanation the result would have been of little interest, but without an elegant result that drew together several mathematical ideas the exploration would have been fruitless.
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Hence I argue that we require a philosophy for school mathematics that adequately reflects both mathematics as a human activity and mathematics as a coherent body of knowledge. One such philosophy is Kitcher’s (1988) mathematical naturalism. Kitcher emphasises the historical development of mathematics as a process that builds on existing knowledge and leads to new knowledge. This occurs through what he terms rational interpractice transitions, where a mathematical practice has five essential components:
a language employed by mathematicians whose practice it is, a set of statements accepted by those mathematicians, a set of questions that they regard as important and currently unsolved, a set of reasonings that they use to justify the statements they accept, and a set of mathematical views embodying their ideas about how mathematics should be done, the ordering of mathematical disciplines, and so forth” (p. 299).
While at first glance Ernest’s (1991, 1998) social constructivism may appear to be a variant of mathematical naturalism, I suggest that there are several important differences. Firstly mathematical naturalism does not deny the existence of an external reality—indeed, it assumes that early mathematical practices were based firmly in a desire to understand the world. Secondly mathematical naturalism emphasises the rationality of mathematics in that mathematical practices change through a rational process characterised by proof and rigour. It explicitly values this rationality, which may be individual or collective, as the process through which epistemic or external ends are met. Truth is seen as “what rational inquiry will produce, in the long run” (p. 314). Thirdly mathematical naturalism does not deny the certainty of mathematics. Rather it suggests that increasing certainty is the ultimate end of mathematical enquiry.