Abstract
According to Spelke and Tsivkin numerals are a linguistic and cognitive bridge between two types of ‘core’ knowledge, that is, subitization of small quantities and approximate representation of large quantities. In this chapter, I go somewhat their way but I also introduce some a priori constraints on what could constitute a bridge. Such constraints are on the ‘design’ of a numeral system and on its use. The starting point is the consideration that numerals like ‘three’ (as well as names of days of the week like ‘Friday’) are nonstandard linguistic items. I propose that their peculiarity is primarily neither a syntactic nor a semantic peculiarity. It is instead in their morphology. Mastering numerals and names for days of the week is assigning them a certain nonstandard morphology, whereby any numeral is mandatorily a non-independent part of a longer sequence. It is hypothesized that this nonstandard morphology is associated with a nonstandard (at least for language) semantics, i.e. map semantics. In a sense, numerals are an artificial language encroached in natural language. The explanatory advantages of the account are discussed and contrasted with Spelke and Tsivkin ‘bridge’ account of the role of numerals in cognition.
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- 1.
A difference is assumed here between sets (abstract entities) and manifolds (concrete entities).
- 2.
Some particular problems posed by ‘eleven’ and ‘twenty’ will not be addressed here. Some of the idiosyncrasies of numeral systems are likely to be independent of the main point of this chapter.
- 3.
Bühler 1934 (Mulligan 1997, p. 201) was probably the first to remark the analogies between these sequences, which he called ‘organizers’. It is interesting to note that names for colours are not on this list, whereas names of musical notes are. It may be that these particular sequences are just an extension of the large class of asymmetric idioms (‘black and white’, ‘up and down’, ‘Tom, Dick and Harry’).
- 4.
Compositionality in the field of numerals is not unrestricted, however, as one cannot form expressions such as ‘twenty-twenty’. However, look carefully at the various possibilities: ‘two-twenty’ is acceptable if used to express the hour of the day.
- 5.
Interference between figural properties and subitization has been documented too (Trick and Pylyshyn 1993).
- 6.
Number cognition is just one example of Spelke and Tsivkin more general hypothesis that natural language bridges different systems of initial knowledge. Another example concerns the integration of space and colour representations (Hermer and Spelke 1996).
- 7.
Hence ‘doggy’ and ‘doghood’ do not qualify as counterexamples. A very strong construal of my claim here is that, actually, ‘three’ is not a word, and the sequence S is.
- 8.
One may use a symbol such as ‘~ …’ to indicate that there is a word, the word replaced for the three dots is a proper part of. ‘Three’ would then be represented morphologically as ‘~ Three’. A way to express this colloquially is to say that ‘three’ is syn-morphological.
- 9.
Some singular terms in natural language are referentially structured in a part-whole fashion that matches the morphological composition: The referent of ‘The county of Exeter’ includes the referent of ‘Exeter’; however, this is not the general case, as the referent of ‘The capital of Italy’ does not include, and is rather included in, the referent of ‘Italy’).
- 10.
Boyer (1991), although this is only a speculation so far as I can ascertain. Another problem concerns the use of numerals in illiterate cultures; one should check where they got their numerals from, for instance, from a nearby culture that in turn incorporated it from written language.
- 11.
It is possible that spoken language incorporates many other nonlinguistic elements, pieces of the external world that we learned to pronounce. Possibly demonstratives are such an incorporation—pronounceable gestures.
- 12.
An old Wittgensteinian lesson.
- 13.
This is anecdotal evidence in need of empirical confirmation.
- 14.
Names of fingers and toes are interesting in themselves as they constitute sequences. Note that pianists have to relearn names for fingers as numerals.
- 15.
Classifying S as a unit is, of course, relatively loose talk. It is not clear that S is a word.
References
Benacerraf P (1965) What numbers could not be. Philos Rev 74:47–73
Boyer CB (1991) A history of mathematics, 2nd edn (revised: Merzbach, UC). Wiley, New York
Bühler K (1934) Sprachtheorie. Fischer, Jena
Casati R, Varzi AC (1999) Parts and places. The MIT Press, Cambridge
Dehaene S (1997) The number sense. Oxford University , Oxford
Dehaene S, Spelke L, Pinel P, Stansecu R, Tsivkin R (1999) Sources of mathematical thinking: behavioral and brain-imaging evidence. Science 284:970–974
Frege G (1883) Grundlagen der Arithmetik, eine logisch-mathematische Untersuchung über den Begriff der Zahl. Nabu Press, Breslau
Gallistel CR (1993) The organization of learning. The MIT Press, Cambridge
Hermer L, Spelke ES (1996) Modularity and development: the case of spatial reorientation. Cognition 61:195–232
Husserl E (1891) Philosophie der Arithmetik. Pfeffer, Halle
Hurford JR (2001) Languages treat 1–4 specially. Mind Lang 16:69–75
Karmiloff-Smith A (1992) Beyond modularity. The MIT Press, Cambridge
Mulligan K (1997) The essence of language: Wittgenstein’s builders and Buhler’s bricks. Revue de Métaphysique et de Morale 2:193–215
Simons P (1982) Number and manifold. In: Smith B (ed) Parts and moments. Studies in logic and formal ontology. Philosophia Verlag, München, pp. 160–199
Spelke ES, Tsivkin S (2001) Initial knowledge and conceptual change: space and number. In: Bowerman M, Levinson S (eds) Language acquisition and conceptual development. Cambridge University Press, Cambridge, 70–97
Trick LM, Pylyshyn ZW (1993) What enumeration studies can show us about spatial attention: evidence for limited capacity preattentive processing. J Exp Psychol Hum Percept Perform 19(2):331–351
Acknowledgments
Thanks to Luca Bonatti, Valentina Gliozzi, Pierre Jacob, Nirmalangshu Mukherji, Marco Panza, Achille Varzi, for useful comments on earlier versions of this chapter. And of course thanks to Kevin Mulligan, who got so many projects on the right track.
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Casati, R. (2014). Numerals and Word Sequences. In: Reboul, A. (eds) Mind, Values, and Metaphysics. Springer, Cham. https://doi.org/10.1007/978-3-319-05146-8_22
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