Abstract
This chapter raises a number of issues from pre-history and history that one mathematics educator considers ‘worthy of mention’ with regard to tools and mathematics. These issues are: tool use in the development of the human species (phylogenesis); tool use in a mathematical culture, ancient Greek mathematics that goes beyond the obvious tools; an example from ancient Indian mathematics that bears some resemblances to Jon’s experimental mathematics described in Chap. 3; the mutual support of hand, mind and artefact in expert use of an abacus; a consideration of a period (sixteenth-century Europe) where there was a rapid advance in the development of mathematical tools.
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Notes
- 1.
Homo sapiens is a specie in the genus Homo in the family Hominidae.
- 2.
Before common era, a term preferred by scholars to BC (but virtually identical in terms of dates).
- 3.
Almost all the mathematicians of ancient Greece were upper class males.
- 4.
Thirteen strictly sequenced books of definitions, postulates, common notions and propositions.
- 5.
Evidence suggests that ancient Greeks also ‘used pebbles for calculations on abaci … but in a marginal role … never at the centre of mathematical activity’ (Netz, 1999, pp. 63–64).
- 6.
Without detracting from the wonder of Greek mathematics, there are mathematical problems with its definitions and proofs. We do not consider these here. The interested reader may consult Netz (1999).
- 7.
- 8.
This makes implicit reference to Proposition I.46. We return to implicit references and expected knowledge in our discussion of the ‘the tool box’ after the proof.
- 9.
This makes implicit reference to Proposition I.31.
- 10.
The complement of a parallelogram would be expected to be known to readers. Line <1> makes implicit reference to Proposition I.43.
- 11.
Implicit reference to Proposition I.36.
- 12.
The gnomon is defined in Definition II.2.
- 13.
‘Netz’ in the following pages refers to ‘Netz (1999)’.
- 14.
I have represented this using a symbol for ‘therefore’ instead of representing this as ‘<1> & <2> → <3>’ as I am far from certain that implication in terms of mathematical logic (suggested by the ‘→’ sign) is how the Greeks understood the relationship between ‘<1>, <2> and <3>’.
- 15.
- 16.
The mathematical community considered also constructed algorithm for computing cube roots but I restrict my focus to square roots in this section.
- 17.
I drop the prefix ‘European’ for the remainder of this section.
- 18.
\( a={ \log}_bc\iff c={b}^a,b>0\ \mathrm{and}\ b\ne 1. \)
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Monaghan, J. (2016). Tools, Human Development and Mathematics. In: Tools and Mathematics. Mathematics Education Library, vol 110. Springer, Cham. https://doi.org/10.1007/978-3-319-02396-0_4
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