Abstract
Mathematicians have often discussed mathematics as a language. Common linguistic categories have analogous mathematical objects. A comparison of linguistic categories and mathematical objects is developed with reference to the early history of mathematics.
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Notes
- 1.
The initial proposal of Lincos most closely approximated the aphorism in its claim that “mathematical expressions and formulae belong to a language different from that of the surrounding context…The syntactical structure of ‘mathematical language’ differs enormously from that of all natural languages” [16, p. 6].
- 2.
For the initial proposal of Astraglossa, see [22]. Hogben presumed the recipients of the messages to be Martians and Astraglossa depends on two-way communication. Hogben most nearly approximated the aphorism by writing that “[n]umber will initially be our common idiom of reciprocal recognition; and astronomy will be the topic of our first factual conversations” [22, p. 260].
- 3.
For example, the order of operations suggests that in 7 × (3 + 2) the addition happens before the multiplication. Independent mathematical statements exist outside of a temporal framework. Without a context established by natural language, the mathematical statements \(6 + 2 = 8\) and 7 × 5 = 35 have no relationship to each other. In contrast, natural languages can often easily suggest a temporal relationship, such as “Six was increased by two to make eight; Seven will be multiplied by five to find thirty-five.”
- 4.
By contrast, the category of Babylonian and Egyptian operations is clopen. Both categories are closed because new Babylonian or Egyptian mathematical texts cannot be created but they are also open because new texts might be discovered or the reconstructions might be reassessed. For example, Plimpton 322 has yielded recent surprises.
- 5.
Popular histories of mathematics often reproduce weakly cited counter-evidence. According to one edition, Chapter 64 of the Book of the Dead tallies 4,601,200 gods [4, p. 164]. Other editions confirm only one million. A grapheme introduced in the Ptolemaic era may express a sense of totality, or cyclical completion [40, p. 12, n. 6], or it may stand for millions, thereby increasing the old grapheme of the god Ḥeḥ to tens of millions [21, p. 280]. Occasionally, decorative hieroglyphs combine the graphemes of large numbers into images which may be read as “a hundred thousand million years” or even “ten million hundred thousand million years” [5, p. 507, s.v. ḥeḥ], but these graphemes are more likely artistic hyperbole than mathematical quantity.
- 6.
The fraction \(\frac{2} {3}\) appears idiosyncratically as the only portion of an Egyptian number which is not easily classified as either an integer or an inverse. (The fraction \(\frac{5} {6}\) is sometimes cited as another example but this fraction is actually a ligature of \(\frac{2} {3}\) and \(\frac{1} {6}\).) This idiosyncrasy may be resolved by morphology. The fraction \(\frac{2} {3}\) was vocalized as rwy, a grammatical dual which literally means “the two parts.” The grapheme is better interpreted as \(\frac{1} {3} + \frac{1} {3}\) rather than a ratio of 2 to 3.
- 7.
The naked inclusion of non-Western languages too often provokes consternation. The phrase may be transliterated as “kō ha sosū da to ı̄mashō” and translated as “Let’s say that is a prime.” The grapheme derives from a pictogram of hands braiding raw silk and the semantic range of the sign includes plain, poor, foundation, and root. The use of kō as a variable is admittedly somewhat contrived for modern Japanese. Probably, a variable from the Roman alphabet would be used, most likely a in slight contrast to the Western preference for p. Mathematics has become an international endeavor.
- 8.
The obvious exception is Indian mathematics which has developed an overabundance of mathematical synonyms. Because Indian mathematical texts are almost exclusively poetic, these variations may be explained by causa metris. However, in many cases, this explanation is facile. To date, a systematic study of whether the variation in vocabulary of Indian mathematics remains a desideratum. Only careful comparison will reveal if these variations indicate regional variations, the temporal development of Sanskrit, or nuances among mathematical objects.
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Acknowledgements
Academics frequently praise books, but scholarly reflections seldom suit the tastes or developmental needs of children. Michel de Montaigne seems apt to inspire future Nietzsches when he declares that “obsession is the wellspring of genius and madness.” In contrast, [10] by Philip Davis allows those who doubt their genius but prize their sanity to pafnuty, that is, “to pursue tangential matters with hobby-like zeal.” Children as young as eleven have been known to incorporate pafnutying into their pedagogical formation. Through this slim volume, Davis tames madness with whimsy and humanizes genius by levity. Should an early exposure to the admixture of language and mathematics distract the student, that is, if the child becomes a recidivist pafnutier, youthful exuberance may be regulated by [11]. If the present argument resembles too much Montaigne’s madness or Davis’ skeptical classicist, it must be realized that despite several introductions and a shared institutional affiliation, this offering did not result from tutelage under Davis. Rather, it is offered as an homage to a past and present inspiration.
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Ross, M.T. (2015). The linguistic status of mathematics. In: Davis, E., Davis, P. (eds) Mathematics, Substance and Surmise. Springer, Cham. https://doi.org/10.1007/978-3-319-21473-3_13
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