# Discrete Infinity

Dan Everett dlevere at ilstu.edu
Wed Jun 11 12:36:17 UTC 2008

```We find indeterminacy in many places. For example, to borrow an
illustration from a draft of someone else's paper, consider the fact
that we could say for any pine tree it is possible to find another
that has more needles. Yet this is not an argument that there is a
pine tree with an infinite number of needles.

There are various ways to derive such facts, generative grammars being
only one, perhaps not the best, way to do so. Note too that a
generative grammar with recursion cannot be proven to derive an
infinite language either.

Further, whether a sentence in any language can be extended or not, is
an empirical question. I think that it is probably the case that there
are languages for which a longest sentence, actually quite short,
might be given.

-- Dan

Quoting "A. Katz" <amnfn at well.com>:

> It seems we are all agreed that the issue isn't infinity. As far as I can
> see, it's indeterminate length. Despite the fact that no person has ever
> uttered an infinitely long utterance -- nor ever will -- and despite the
> fact that the inventory of actually spoken sentences throughout the
> history of any language is also a finite number, the upper bound on
> whatever that finite number is is indeterminate. That is what gives us the
> freedom to say something original. Granted, original sentences are rare,
> but the possibility of having them crop up is a very big deal. It allows
> us to express new ideas, if and when they occur to us.
>
>
> Best,
>
>      --Aya
>
> On Wed, 11 Jun 2008 dlevere at ilstu.edu wrote:
>
>> Just to clarify what I said earlier.
>>
>> It isn't clear what 'discrete' adds to our understanding of the nature
>> of language since we already have to have linguistic units like words.
>> Since no one would utter half a word, based on what a SIGN is, we
>> don't need the mathematical concept of 'discrete' so far as I can
>> tell. Saussure already got this fact for us.
>>
>> And since no language can be proved to be infinite nor is infinitude a
>> necessary nor a sufficient condition for any language, 'infinity' also
>> plays no role in understanding language. Now, it is true that the
>> shortest grammars that describe many/all natural languages might
>> themselves generate infinite languages, but that is an artifact of the
>> grammar, not the language the grammar is describing. Recent work at
>> MIT (Brain and Cognitive Sciences) by Josh Tenenbaum and his lab
>> (http://web.mit.edu/cocosci/josh.html) has made some excellent
>> progress in offering ways to select among grammars based on their
>> parsimony. And parsimony is the big attraction to infinitude - but it
>> is not a fact about languages so far as I can tell, just the shortest
>> way to describe them.
>>
>> If I am correct that Piraha and other languages are finite, then
>> discrete infinity is not only unnecessary, it makes the wrong
>> predictions.
>>
>> So when someone says that our main task in describing language is to
>> capture its 'discrete infinity', then, to quote Paul Feyerabend's
>> remarks on people who take their views seriously, 'smell a rat'.
>>
>> Dan
>>
>> Quoting Daniel Everett <dlevere at ilstu.edu>:
>>
>> > Fritz,
>> >
>> > If anyone had a clear idea of what discrete infinity was or what work
>> > it does at all, then I might see your point.
>> >
>> > See the special recursion issue to appear from The Linguistic Review,
>> > based on papers from the MPI-ISU conference that I organized, or anyone
>> > of a number of recent papers by Geoff Pullum for some extremely
>> > interesting criticisms of linguists' ineptness in the use of terms like
>> > infinity  and discrete in such contexts. Linguists by and large should
>> > be as wary of using math to justify analyses as Martin H advises they
>> > should be to use 'cognitive'.
>> >
>> > Dan
>> >
>> >
>> > On Jun 10, 2008, at 9:24 PM, Frederick J Newmeyer wrote:
>> >
>> >> I would think that for any semiotic system involving discrete
>> >> infinity, the existence of rules (schemas, constructions) would be
>> >> the null hypothesis.
>> >>
>> >> I don't pretend to have read all of the literature on formulaic
>> >> language. But my impression is that those who put such language on
>> >> centre stage (1) focus almost exclusively on language production
>> >> and all but ignore comprehension and (2) show no interest at all in
>> >>  language users' ability to make judgments of well-formedness of
>> >> sentences that they have never heard. It seems self-evident to me
>> >> that once comprehension and judgment data are brought into the
>> >> picture, the need for rules (schemas, constructions) becomes
>> >> indispensable.
>> >>
>> >> Let me stress that I am NOT offering an argument for 'innateness'
>> >> here. I am not even offering an argument for generative grammar, as
>> >>  opposed to, say, cognitive grammar or construction grammar. Just
>> >> an  argument for rules (schemas, constructions).
>> >>
>> >> --fritz
>> >>
>> >> Frederick J. Newmeyer
>> >> Professor Emeritus, University of Washington
>> >> Adjunct Professor, University of British Columbia and Simon Fraser
>> >> University
>> >>
>> >> On Tue, 10 Jun 2008, Martin Haspelmath wrote:
>> >>
>> >>> It seems to me that Fritz Newmeyer's appeal to the Rule-List
>> >>> Fallacy in the context of the argument about formulaic language
>> >>> overlooks a crucial asymmetry between rules and lists:
>> >>>
>> >>> While lists are a necessary component of all semiotic systems,
>> >>> rules are not. All languages must at least have lists of
>> >>> morphemes, and then in addition they may have rules. But the
>> >>> burden of proof is on those who want to claim that they have rules
>> >>>  (or schemas, or constructions). In general, the evidence for
>> >>> rules  has been considered overwhelming (in all languages), so
>> >>> almost  everyone accepts them.
>> >>>
>> >>> Now I think Fritz's argument doesn't go through: If one could show
>> >>>  that it is in fact possible to explain speakers' behaviour by
>> >>> claiming that their knowledge of language consists of a simple
>> >>> list of morphemes (or formulas), then this would indeed be a
>> >>> powerful argument against the existence of rules. In other words,
>> >>> the null hypothesis should be that languages have no rules, and if
>> >>>  not enough evidence can be found to reject this hypothesis, we
>> >>> should assume that they don't.
>> >>>
>> >>> Notice that this doesn't work the other way round: The null
>> >>> hypothesis cannot be that languages have no lists, but only rules
>> >>> -- languages must have lists. So if one discovers rules, this does
>> >>>  not mean that the same phenomena are not also stored as lists.
>> >>> The  Rule-List Fallacy is unidirectional.
>> >>>
>> >>> But while I think that this particular argument is invalid, Sandy
>> >>> Thompson and Paul Hopper will need to do a lot more to convince
>> >>> linguists that no rules (or schemas, or constructions) are needed
>> >>> to explain speaker behaviour. Strictly speaking, they are
>> >>> defending the null hypothesis, but in actual practice, almost all
>> >>> linguists (regardless of their ideological preferences) find that
>> >>> they need rules for their work.
>> >>>
>> >>> Martin Haspelmath
>> >>>
>> >>> Frederick J Newmeyer wrote:
>> >>>> Let me start by calling attention to what Ron Langacker has
>> >>>> called the 'Rule-List Fallacy'. Ron noted, completely correctly
>> >>>> in my opinion, that it was a fallacy to assume that lists have to
>> >>>>  be be excised from the grammar of a language if rules that
>> >>>> subsume them can be established. The converse of this fallacy is
>> >>>> equally fallacious: that rules have to be be excised from the
>> >>>> grammar of a language if lists can be established. Even if it
>> >>>> were the case that a huge percentage of language users' output
>> >>>> could be characterized by lists (formulas, fragments, etc.), that
>> >>>>  would not exclude their also have a grammar composed of rules
>> >>>> (or  their notional equivalents) that allow hearers to analyze
>> >>>> unfamiliar collocations and assign to them structure and meaning.
>> >>> --
>> >>> Martin Haspelmath (haspelmath at eva.mpg.de)
>> >>> Max-Planck-Institut fuer evolutionaere Anthropologie, Deutscher
>> >>> Platz 6	D-04103 Leipzig Tel. (MPI) +49-341-3550 307, (priv.)
>> >>> +49-341-980 1616
>> >>>
>> >>> Glottopedia - the free encyclopedia of linguistics
>> >>> (http://www.glottopedia.org)
>> >>>
>> >>>
>> >>>
>> >>>
>> >>>
>> >>>
>> >>
>> >>
>>
>>
>>
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>>
>>
>>
>

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