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Indistinguishable from Magic: Computation is Cognitive Technology

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Abstract

This paper explains how mathematical computation can be constructed from weaker recursive patterns typical of natural languages. A thought experiment is used to describe the formalization of computational rules, or arithmetical axioms, using only orally-based natural language capabilities, and motivated by two accomplishments of ancient Indian mathematics and linguistics. One accomplishment is the expression of positional value using versified Sanskrit number words in addition to orthodox inscribed numerals. The second is Pāṇini’s invention, around the fifth century BCE, of a formal grammar for spoken Sanskrit, expressed in oral verse extending ordinary Sanskrit, and using recursive methods rediscovered in the twentieth century. The Sanskrit positional number compounds and Pāṇini’s formal system are construed as linguistic grammaticalizations relying on tacit cognitive models of symbolic form. The thought experiment shows that universal computation can be constructed from natural language structure and skills, and shows why intentional capabilities needed for language use play a role in computation across all media. The evolution of writing and positional number systems in Mesopotamia is used to transfer the thought experiment of “oral arithmetic” to inscribed computation. The thought experiment and historical evidence combine to show how and why mathematical computation is a cognitive technology extending generic symbolic skills associated with language structure, usage, and change.

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Notes

  1. Fowler (1999, chap. 1) discusses Greek mathematics as “non-arithmetical” and (Unguru 1991; Fowler 1994) debate the scope of Greek induction, a close relative of recursion.

  2. For example, Egyptian mathematics used an additive number system which, through a clever trick, could also be used for reasonably efficient multiplications. But that construction, while effective and correct, is not efficient enough for much further algorithmic design. Similarly, one can use high school Roman numerals for multiplication, but they quickly become difficult to manipulate further. The great historical and cognitive solution here is positional value, whose compact linguistic expression in ancient India plays a key role below. See also note 25 below.

  3. “It is in this sense than an arbitrary Turing machine, or an unrestricted rewriting system, is too unstructured to serve as a grammar. By imposing further conditions on the grammatical rules, we arrive at systems that have more linguistic interest but less generative power”(Chomsky 1963, 359). Hence natural language recursion, however complex, is still computationally weak; see also (Pullum and Scholz 2005; Chomsky 1980, 123).

  4. I focus on the transition from additive to multiplicative algorithms because: (1) additive number systems, while not universal across languages, are common and easy to construct in many grammars, with number words coordinating, e.g., body-part counting and other one–one enumeration behaviors, possibly sufficient for bounded multiplication; (2) in formal models of arithmetic, addition and multiplication (but not addition alone) are sufficient to represent universal compution; (3) with multiplication and universal computation, intensional phenomena occur in the formation of consistency statements (Boolos and Jeffrey 1979, 186), and these are a symptom of, but not identical with, intentionality (Searle 1983, 24).

  5. Boas and Powell (1911). More recently, (Everett 2005) controversially argues that the Amazonian Pirahã language lacks recursive syntax in ways challenging Chomsky’s claim that recursion is a linguistic universal; Everett also conjectures that a stringently local and finitist worldview makes some abstractions and syntactic forms mostly unnecessary. My approach is a complement: given certain types of linguistic recursion, whether universal or not, then one can construct mathematial recursion from it as discussed below. As with Everett, the material and cognitive setting is all important to the development of grammatical form.

  6. See (Dehaene 1997, chap. 9) on cognition and modern mathematics; the present paper addresses some of the issues raised there.

  7. See (Clark 1996; Tomasello 1999, 2003) on language use and the coordination of intentionality.

  8. Number systems abound in history which extend counting behaviors, especially one–one enumerations, through number words and their productive rules. But not found are non-written linguistic examples with much computing power beyond simple addition or very limited multiplication, and so these systems don’t reveal much about cognition for modern computation and mathematics.

  9. Gentner and Goldin-Meadow (2003) revisit the Sapir-Whorf hypothesis, not generally thought of as including writing and computation (Whorf 1956). While the thought experiment in the text shows that, in principle, linguistic change can lead to language in which all computations are possible, in reality the change relies on our perceptions of inscribed computations. So for computation, the increase in conceptual power depends critically on changing media to realize the more intricate symbolic apparatus.

  10. See (Slobin 2003) on “thinking for speaking”; (Dehaene 1997, 102) on counting skills for native Chinese speakers; and (Levinson 2003) on spatial marking.

  11. See (Bloom 1994) on the transfer of recursion across cognitive domains; my approach focuses on external media as perceived and leveraged by fixed cognitive capacities.

  12. For example, in logic, “alphabet” and “rewrite” should be metaphors whose mathematical content has nothing to do with inscription per se, just as Plato advised that geometry “…is in direct contradiction with the language employed in it by its adeptsthey speak as if they were doing something [practical]… their talk is of squaring and applying and adding and the like, whereas in fact the real object of the study is pure knowledge” (Republic 518d, 527). Difficult though, is understanding what an abstract discrete symbol is outside of any language or semiotic system at all; see “Grammaticalization in Script” below.

  13. While modern generative linguistics borrowed formalist ideas from mathematical logic, ancient Indian linguists mostly developed them directly, building on basic ideas of recursive system construction developed to exactly describe generatively patterned rituals, with portions of the early Vedas known as “ritual manuals” (Renou 1941, Staal 1990). Included here are the earliest phonological theories and the segmentation of continuous recitation into discrete units (saṃhitā vs. padapāṭha) for analytical purposes (Staal 2006, 77). On expressions of generality and the potential infinity of language generated by finite means in Indian grammatical theory, see (Staal 1990, 89).

  14. “These sūtras are like nothing so much as the rules in a comptuational grammar of a modern language, such as might be used in a machine translation system: without any musical or ritual element, they apply according to abstract formal principles. This is not a metaphor, or anachronistic interpretation of Sanskrit grammar, but a straightforward description of the working of the sūtras in Pāṇini’s system”(Ostler 2005, 181). This is a slight overstatement because of initializing kāraka rules, requiring user-based assignments of categories of “agent,” “patient,” etc.; see (Gillon 2007).

  15. For introductions to Pāṇini’s grammar see (Gillon 2007; Sharma 1987) On the evolution of metalinguistic concepts in India see (Staal 1975).

  16. It was essential that Pāṇini’s artificial langauge extend Sanskrit in a natural way for its use would otherwise be seen as polluting, or at least vulgar, given the privledged status given language (vac or bráhman) in Vedic culture. Pāṇini’s grammar was meant to ensure exact oral reproduction across generations because the linguistic expression itself, when a correct copy of the original, was sacred, the voices of men speaking the language of gods. But pragmatically, Sanskrit was also intended as an efficient and accurate means of oral communication across a huge land, and the grammar provided needed consistency. Hence the grammar is descriptive but was taken as prescriptive for a combination of metaphysical, pragmatic-communicative, and socio-political reasons (Staal 1995).

  17. Positional value can be defined by a schema for concatenated symbols a 1a n+1 such as pos n : a 1a n+1 = (10 × a 1a n ) + a n+1. These equations can be defined in formal arithmetics for addition and multiplication, as can a single master formula Pos ( n, S) from which all pos n can be derived for a finite symbol set S. In fact, addition alone (e.g. the complete and decidable formal theory Presburger arithmetic) is sufficient to individually define each pos n , but a formula Pos ( n, S) involves general multiplication and so cannot be defined using addition alone. Simple formalisms including addition and general multiplication are sufficient for universal computation and metasymbolic operations generally, leading to incompleteness and undecidability results, including the intensional unprovability of consistency (cf. note 4 above). The two formal theory classes, of additive and multiplicative arithmetics, are useful correlates for the changes L → L′ discussed in the text (Kadvany 2007).

  18. The problem of automating number units was expressed by Archimedes in The Sand Reckoner, a tour de force in which he used a geometrical model to estimate the number of “atoms” in the universe as a myriad-myriad units of the myriad-myriad-th order of the myriad-myriad-th period, or [(108)10^8]10^8 (Dijksterhuis 1938); a myriad is 104 and a myriad-myriad is 108. Cognitively, positional value, not discovered by Archimedes or other Greek mathematical giants, is a principal means by which algorithms using multiplication and exponentiation are easily formed, as described by Fowler in note 25 below. On “automation” as a product of grammaticalization see (Bybee 2003).

  19. See (Jackendoff 2002) on the generative power of grammatical constructions making limited assumptions about syntactic structure; hence construction grammar does not imply “non-generative.”

  20. See (Goody 1987, chap. 4) on the likely role of writing in the design of Pāṇini’s complex matrix of phonemes, the Śivasūtras, and many other observations on the difficulties of expressing formal concepts (lists, tables, multiplication) in oral cultures. For other indicators of written influences on Pāṇini see (Datta and Singh 1935, I 18, 33).

  21. The bridges existing today between mathematics, generative linguistics, and computation did not quite exist for Indian linguistics and mathematics, even as each exploited aspects of algorithmic method unified in the twentieth century. In India, linguists used powerful algorithmic formalisms, but applied to language, not mathematical algorithms; the formalisms were not needed by mathematicians, who nonetheless made algorithmic design a key heuristic tool and method of expression, as opposed to proof structure. Shared features of linguistics and mathematics included the use of Sanskrit sūtras to describe algorithms or linguistic rules; and Sanskrit number words to codify positional notation using compounds ratified, in turn, by Pāṇini’s linguistic rules.

  22. See note 7 above, as well as (Baron-Cohen 1997) on mindreading and triadic representation. With others, Baron-Cohen considers the inability to process triadic representations a critical factor in some forms of autism, an important observable correlate of the cognitive abstractions.

  23. (Karmiloff-Smith 1992: 132). See also (Leslie 1987, 1994) on pretend play and symbolic decoupling. As with Baron-Cohen (note 22), Leslie’s modularist framework is not critical for many basic observations.

  24. Chomsky made Wilhelm von Humboldt the first hero of the linguistic infinite (Chomsky 1965, v). But Johann von Herder was as visionary about the mind as Humboldt was about language, speculating that language is the product of what we today identify as reflective intentional capacities (von Herder 1772, esp. 86–87). Intentionality does not start with Franz Brentano or Edmund Husserl, but has a longer history associated directly with the linked philosophies of language and mind (Aarsleff 1982).

  25. David Fowler writes of Simon Stevin’s 1585 introduction to positional notation De Thiende (“The Tenth”) that “Decimal fractions gave a new fluency to arithmetic which permitted, perhaps for the first time [sic: in Europe] the feelings that all such calculations could now be taken for granted, and this paved the way to the next stage, their abstraction into symbolic algebra…the people who contribued to theis development—principally Stevin himself, Viète, and Harriot—were themselves calculators…This confidence in decimal arithmetic still lies, I believe, behind our basic intuitions, even today, underlying the real numbers, even though it is a delusion…” (Fowler 1999, 406; emphasis added). By “delusion” Folwer means that generating decimal expansions for answers to simple arithmetic operations can be very complicated, e.g. because of very slow convergence, such as occurs in many power series expansions for pi.

  26. Wilfried Sieg and Saul Kripke have argued that the Church-Turing thesis can be deduced as a theorem from axioms for relevant combinatorial relations intended to represent what they take to be machine, rather than human, computations. The Church-Turing thesis, they argue, therefore does not have to be conceived as an informal thought-experiment about human computers as devised by Turing (Sieg 2008; Kripke 2000). But such theorems rely on axiom systems which are further computational idioms, expressed in artificial languages requiring apparatus for their construction as discussed in the text. Frege also overreached with his wholesale rejection of “psychologism” in mathematics (Frege 1884), though of course he lacked a modern view of how languages, including his own artificial language for predicate logic, rely on specialized cognitive and symbolic skills. John Searle has pointed out the intentional role in computation generally (Searle 1992, chap. 9), also arguing that any process can be identified as “computation” by intentional fiat, an observation he takes as superceding his older “Chinese Room” thought-experiment meant to demonstrate the intrinsically intentional status of language use. The text supports those views by emphasizing the particular symbolic skills used in the formation and use of artificial languages.

  27. Powell (1976) describes pre-Babylonian number words using multiplicative units like named powers of ten in India. On the evolution of numbers, cuneiform signs, and positional notation see (Høyrup 1994; Nissen et al. 1993). On measurement and multiplicative units see (Krantz et al. 1971, Ch. 10).

  28. “The ‘great invention’ [of writing] was almost certainly the prehistoric move from a token-iterative to an ‘emblem-slotting’ [a generic number sign combined with a separate category] for recording numerical information…Slotting is a structural technique we now regard as intrinsic to language; and nowhere more typically than in the way languages deal with counting.” (Harris 1986, 145). On slotting see also (Tomasello 2003, 122–126).

  29. See note 12 above and text.

  30. Stimulated by Donald Knuth’s TeX typesetting language, (Hofstader 1985) argues that no fundamental criteria characterize variations across all font designs, say for the letter “a”. Thus graphemes are not better defined than phonemes (Coulmas 2003, 204).

  31. See (Jakobson 1990, 240) on this synthetic duality in phonemes, and (Tomasello 2003, chap. 5) and references there on similar form-function dependencies in syntactical analysis.

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Kadvany, J. Indistinguishable from Magic: Computation is Cognitive Technology. Minds & Machines 20, 119–143 (2010). https://doi.org/10.1007/s11023-010-9185-z

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