Neither
languages nor linguistic competences are usefully said to be ‘infinitary’
It
has often been said that a central problem/question of theoretical linguistics
and of the philosophy of language is something like the following: (Qu.):
How can finite beings master infinite systems, such as natural
languages? [1] But it is
often radically unclear what is meant by the terms ‘finite’ and ‘infinite’ in
this context; it is unclear how they are supposed to play a role either in
stating or in resolving a problem. It is my view that the question raised above
contains a deep and probably irresolvable unclarity, and thus contains at least
one important mistake.
I
will argue for the following: insofar as we can assign / give any clarity to
the concepts ‘natural language’ and ‘infinite’, it is at best unclear how and whether we can intelligibly speak of
natural languages as infinitary systems.
Idealising
slightly, by “a language” I understand a notation [2] which can (at least in principle) be employed for
communicative purposes, and so allows the composition of meaningful units (i.e.
sentences). Thus correlated with a language (e.g. ‘English’) is the set of
sentences which it facilitates - some prefer to say that a language is
constituted by this set. By ‘infinite’ I understand something qualitatively
different from mere very great or extraordinary size or duration - an infinite
system or quantity must be unbounded, without a point which could possibly be
the termination of the system/quantity.[3] Insofar as there is one, the most natural
paradigm case of infinitude is simply the expansion of the natural numbers:viz. 1,2,3,4,5... . (Though the dots at
the end ought not to be read as the dots of abbreviation. The sense in which we
understand the infinitude of the natural numbers, in prose rather than in the
course of (e.g.) some technical-mathematical manipulation of some transfinites,
is only the sense in which we understand that there is no provision for - no ‘room’ for - a last term in the series
of natural numbers. Or again, the sense in which any series of natural numbers
which we choose to develop will have a highest term only ‘arbitrarily’, not as
a matter of absolute numerical principle.[4] )
But
each and every sentence of a language must be composed out of a finite number
of (finite) compositional units (viz.
words of the language, punctuation marks, etc.).
And each and every sentence must be finitely long - we cannot even in principle
make sense of the concept of an ‘infinitely long sentence’, because it would
never be clear that it was a sentence, let alone what sentence it was (This follows from
the remark about the infinite, above; we shall return to it later).[5]
Now,
it has of course been observed that no human ever literally exhibits or performs an infinite linguistic competence. But the above
paragraphs entail that we can say (further and more strongly than this) that -
no matter how you permute or count the sentences which are correlated with /
constitute a language - if they are finitely long and built up from a finite
number of elements then they are
themselves finite in number.[6] So we have
as yet no infinite linguistic system which one should need to have
mastery/competence in.[7]
This
appears to be a valid argument against the rightness, the intelligibility, of
the ‘central question’ (‘(Qu.)’), mentioned above.
I
can see two objections to this argument. One is the obvious one, classically
cited by linguists and logicians, and, to some
degree, surely correct - that we can repeat words in a sequence, and thus that
there is a potential infinity of finite (and grammatical) sequences. I will
respond critically to this in a moment. First though, I want to bracket this
objection, and deal with the only other telling objection-schema I can imagine
to the argument I have given, should
this ‘repeatability of words’ objection prove to some degree ill-founded. The
other telling objection would run as
follows: “But is this not a legitimate English
sentence: “10000001! is a very large
number indeed”? Surely it is, and an infinite number of such supra-mathematical
sentences may be formulated, by means of inserting in turn all the numbers of a
denumerable set. So the assumption that there is a finite number of elements
from which sentences may be constructed must be false. Does this not show,
then, that languages/linguistic systems must be infinite, by
non-controversially pointing to the nesting of the sentences of mathematics
within those of (e.g.) English?” (One could also run this objection simply by
treating mathematical ‘sentences’ directly as sentences of English.)
A
full treatment of this objection would take us deep into (particularly) the
philosophy of mathematics. I will restrict myself to making two points about
it.
To
begin with, the objection has only limited scope in application, in that if
language is infinite only in virtue
of its including series which are designed/defined
such that they are infinite, then one would have at least to agree that, absent
the creation/discovery of such series, language and linguistic ability would be
(and at one point surely were)
reassuringly non-infinitary. (It might be objected here that, because of the
numbers always being there to be discovered - or because of the ability always
being at least latently there, waiting on its historical destiny - linguistic
ability can always be said to have been infinitary. But this is hyperbolical Platonism. Such latent
‘there-ness’ is simply a very imaginative abstract retrospective construct. We
might as well say that the concept of ‘xylophone’ is innate; or that we have
the (latent) ability to reproduce asexually and spontaneously.[8] ) Note that one cannot simply appeal to the role
of terms such as ‘infinity’ or ‘aleph0’ in our language - such terms have a fairly
well-defined and fairly ordinary role (comparable to that of words such as
‘God’ or ‘the Universe’ or, come to that, ‘differance’
), and do not in and of themselves make any progress toward generating an
infinitary linguistic system.[9] One must suppose that it is the possibility of
generating endless specific sentences
about specific numbers that entails that our language be infinite.
Relatedly,
and more tellingly, this above-canvassed objection of course arrives at the
infinitary nature of language only by
building this in through having a language be built out of an infinity of
elements (i.e. 1, 10, 101, etc. And surely we do not wish to put all the weight
of language’s supposed infinitude upon the mere possibility of endlessly adding
(e.g.) “0” or “1” to whatever number one comes up with?). Traditionally, the
conundrum with which this paper opened has been posed as a puzzle about ‘finite
beings’ manipulating a finite array of symbols and yet generating and having an
ability ranging over an infinitary symbolic system. So my initial argument
would, if valid but not sound, at least necessitate a location of the puzzle at
a different level than has been usually supposed - the puzzle would become one
of how it is possible to generate an infinite array of possible linguistic
elements, of symbols. But again, that question we already know the
(rather trivial) ‘answer’ to, insofar as there one: simply develop a
recursively denoted and endlessly inscriptible series - viz. the natural numbers.[10]
Once again, then, we find that it could only
be in virtue of having developed notations which specifically - definitionally
- allow for counting to infinity (as opposed, for instance, to a system of
numerals amounting to ‘1,2,3,4,5,Many’) that our linguistic ability could
justly be conceived of as infinitary. And it is vital here to bear in mind
Wittgenstein’s grammatical point that even an ability conceived thus as
infinitary - through an infinitude of possible linguistic elements - would not
thereby realize an infinite series,
nor exhibit an infinitary competence, but only satisfy a norm of
representation:
“If
1 is divided by 3 there is no such thing as an infinite series of 3s. There is
a law that 1 divided by 3 is 0.3 recurring. We confuse the infinite possibility
of writing 3s with 3s written down. Similarly, there is an infinite possibility
of constructing points on a line, but a line is not therefore made up of
points.” [11]
So,
having dealt with the objection from mathematical language, let us consider the
more familiar objection based on sentences like “Jean was not not a good
person” and “It was sunny and it was the 4th of July and things had never been
better and yet everyone was unhappy”. Through structures such as these - the
iteration of logical connectives, either with or without the addition of
further clauses/sentences - it is held that, though sentences must be finite,
and built up from a finite pool of linguistic elements, some members of that
pool can be repeated indefinitely many
times, and that such potentially-boundless repetition entails the
conclusion that natural languages, and our linguistic capacity, are infinitary.[12] This
objection potentially invalidates the initial argument I gave, for it has the
consequence that a finite number of linguistic elements need not entail that
the total number of generatible sentences is itself finite. I did, it seems,
omit to deal with the - relevant - possibility of the same word or phrase being
rightly iterable.
But the postulation of (the need for)
infinitary linguistic competence (which, as already mentioned, cannot be
matched by performance) may rest upon confusions related to those excavated
above. I wish to challenge the competence vs.
performance distinction, by means of challenging the tenability of the
apparently-flawless method of constructing or generating infinitely many
indefinitely long sentences which is the linguist’s main means of generating
it. I wish to do this by means of invoking some considerations recently adduced
by Peter Winch in discussing the iterability of negation.[13]
Winch’s
argument comes down to this: One of the key insights prompting Wittgenstein’s
change of philosophical heart beginning in the late 20s was that the logical
connectives are not machines running on rails to infinity - in short, that they
do not contain within them an eternally iterable meaning. It does not follow from the meaning of ‘not’ that
two ‘nots’ cancel each other out in what we are pleased to call
‘double-negation’ (cf. PER, p.126).
Thus
one has to ask how relevant to natural language is the idealisation made when
one imagines that “not” can be understood, in relation to questions of
combinatoriality and generativity, as having as its essential properties those
which we imagine are possessed by the logical symbol for negation. In
particular, if confronted (outside linguistics textbooks) with a long sentence
involving a whole series of ‘“not”s’ counterposed to one another, perhaps
thirty or more, should we presume that this sentence must rightly be understood
in one way or another dependent solely upon whether or not the number of
‘“not”s’ in a row is odd or even? Would it not be to lay down a new rule, if one were to state that a
sentence such as “The return of a Bush as President is not not not not not not
not not not not not not not not not not not not not not not not not not not not
not not not not to be tolerated!” must be understood (if uttered sincerely) as
insisting that we ought to tolerate (a) George Bush’s becoming President
(‘again’)? [14] I think that, if not for this sentence then at least for one somewhat longer (!), it
would be to lay down a new and contestable rule,[15] even if the utterer of the sentence in question
were aware of the exact number of ‘“not”s’ contained in her utterance. One is
reminded here of an argument that Kripke (in criticizing Benacerraf on what the
numbers are) has made concerning the presentation of the natural numbers.
Crudely summarized the argument has the following phenomenological and
rhetorical form:[16] If we were
to be presented with what are purportedly the natural numbers in a form so
peculiar (e.g. in some very large and arcane base) that no human being were
able to recognise the series as the
numbers, would there remain any grounds for saying that the series still might be the numbers? The analogy would then
go as follows: If one is presented with an unrecognisable
sentence - that is, a purported-sentence whose syntax and thus whose semantics
is unsurveyable - does one have any reasonable/plausible grounds for thinking
that nevertheless this could be a sentence of English? (There will be
borderline cases here, but they arguably mark off, in ways with which we are
fairly familiar, the (vague)
frontiers of what we may be pleased to call a finite system, so long as we do
not have in mind a substantive contrast with a conception of an illusory -
infinite - system)
Analogous
points will surely hold for the other iterative devices of natural languages;
that is, for the other logical connectives (besides “not”) too, including even
the innocuous “and”. I.e. Without endorsing some weird and impossible
rule-scepticism, I think one can plainly endorse the suggestion that the
meaning (indeed, the very grammaticality)
of many very long strings may be largely or completely unsettled prior to their
actual consideration by human beings who use or encounter them. If you string
enough phrases and sentences together using “and”, the result may be
unrecognisable by any master of natural language as a sentence. And without
endorsing some stupid verificationism, I think that it is plain that one can
back the view that if no master of a language can recognise some string as in
and of that language, then that string is simply not (at least, as yet) part of that language. Language, to be language, is
and must be both usable and to a very great degree surveyable.[17]
This
results in a substantive counter-argument to the notion that one could in
principle formulate endlessly many grammatical strings from a finite number of
linguistic units. It re-establishes, I believe, the validity of the argument
with which this paper commenced.
There
is an interesting contrast between the argument just given and the assumption
of Charles Sayward, in a piece whose perspective is otherwise fairly conducive
to mine, that “[a] string of 43 trillion occurences of ‘it is not the case
that’ followed by ‘is raining’...is an understandable string. It means that it
is raining.” Sayward ignores that the computation/parsing of any such string
would require new proof procedures, which we should not necessarily be bound to
accept (for in any real situation there would be the question of whether there
were actually exactly 43 trillion); but even if we accept his far-fetched stipulation, the key point is that the
semantics of radically unfamiliar or weird strings is in most cases up for grabs; unless perhaps they are
clearly metaphorical, which hardly helps (because compare Davidson, Guetti etc. on new metaphors). This imagined
string has but little prior claim to be part of the language. Allowing that it is hampers Sayward’s argument toward
the conclusion that “English is indeed finite”, and leads him to commit some
rather desperate and arbitrary finitistic maneouvring [Cf. the Appendices, below].
The
important thing to notice about my Winchian argument is, again, that it does not depend upon finitistic premises, in
the manner in which the considerations cited in some of the footnotes to this
essay might be argued to do so. It is
a logical argument, in the sense of that word that we find in the later
Wittgenstein: it is not physiological or psychological in nature, and is only
(briefly) phenomenological as a means
to addressing logical issues. To make the Winchian move here is to deny that
the grammar of natural languages is endlessly iterable in the way in which most
linguists and philosophers have in recent times supposed that it is.
In
conclusion: simple mathematical infinity - the infinitude of the natural
numbers - provides us (at best) with the only clear notion of infinitude that
we have, and, (excluding in a certain attenuated sense mathematics itself), languages simply are not infinitary systems.[18] In the (limited) sense in which it is possible to
envisage an infinite series of natural numbers (or of “3”s or “2”s), the sense
in which there need be no last term, there is no corresponding series of
sentences in any natural language... for such sentences, sentences that (could)
make a grammatical difference, must be finitely long and finite (though incalculably large) in number.[19] Humans -
‘finite’ beings [20] - need not
be hypothesized as having a linguistic capacity of infinite scope - only,
perhaps, of ‘very large’ scope (though large compared to what, we still ought to ask. Can one language be larger than another? Does the linguistic
ability of a human outshine in magnitude
that of a chimp? Saying “only very large” risks buying into the comparison of
the finite and ‘the infinite’ that Wittgenstein so strongly warned us against;
cf. the Appendices, below. The ‘central problem’ with which this paper began is simply not a problem at all.
Given
this, one might ask what the implications are for linguistics and philosophy? I
would expect them to be confined exclusively to the theoretical level; but
there they might be substantial. For instance, here is one thought-experiment
which, ironically enough, perhaps becomes possible
conditional upon my argument being unrefuted: Imagine a human being whose
memory were exceptionally large and whose life were exceedingly long; she might
actually know all the decidably
grammatical strings of a language. (I entertain this ‘thought experiment’ on
purely ad hominem grounds; as,
speaking for myself, it trades on a deeply confused picture of how and what
language might be. That is, it reifies language, while, as Nielsen, Guetti, and
the recent Davidson among others have pointed out, ‘language’ is in its action, in its meaningful use.
To postulate it even hypothetically as wholly ‘memorized’ is simply absurd; for
what would be memorized would be at best idling ‘senses’, not meanings or uses
at all. The tired old picture of a ‘box’ of some kind explaining one’s
linguistic abilities as they are actualized (and we should not be trapped into
staticizing and reifying ‘ability’ either, as I have been trying to show - and
see Appendix 1, below) must be discarded). What
then would become of the supposed ‘creative’ aspect of language-use so beloved
of generative grammarians?[21]
Appendix 1: Some
additional points concerning Wittgenstein on the infinite.
Some
of this argument may have been too severe. For it may hae veered close to
dogmatic, or psychlogistic, or finitistic argument, where what is required is a
clear understanding of the logic of ‘infinity’, ‘language’, etc. .[22]
What
are the central such points that W. makes? We have mentioned most of them. Here
there are (re-)summarized:
1) One must not confuse ‘finite totality’ with
‘infinite system’. In Shanker’s words:
“The great problem with transfinite set theory lies
in its assumption that ‘ we can understand the meaning of a class without
knowing whether the class is finite or infinite, that that is something we
establish only later...A correct symbolism has to reproduce an infinite class
in a completely different way from a finite one. Finiteness and infinity must
be obvious from its syntax. In a correct language there must not even be a
temptation of raising the question whether a class is finite or infinite.’ The
crux of the issue, therefore, is that ‘
“Infinite” is not a quantity. The word “infinite” has a different syntax
from a number word”(WWK, 228).[23]
Thus referring to transfinite numbers
is liable to confuse us, as it may have confused Cantor (who was extremely
willing to be so confused, arguably). “Wittgenstein did not dispute the wisdom
of [Cantor’s] claim, provided that it is intended to suggest that the
transfinite numbers represent an extension of the concept of number simpliciter, in which case it reveals
an intuitive awareness of the family-resemblance character of number and the autonomy of individual
number-systems.”[24]
The
“logico-syntactical distinction between
finite class/totality and infinite
class/boundless series”[25] is the core of W.’s remarks on “the infinite”.
2) Correlated with this is the
distinction between correlation and calculation, or existence. Thus one can
correlate 1 with 1 , 2 with 4, 3 with 9, etc., but this gets us no nearer being
able to assert things of transfinites with ‘different cardinalities’, to being
able to compare their ‘magnitudes’, to being able to count ‘them’, still less
to being able to count/calculate with
them.
3) As for infinitude vis-a-vis language, Wittgenstein never directly addresses the topic. His
influence has been strongly against the more rampant leanings and
pronouncements of the generative grammarians, but, as stated at the opening of
this Appendix, one needs to take care:
i)
Note his anti-revisionism, which is seen at or near its extreme in debate with
Turing in LFM. Here, as well as often elsewhere, he is very careful not to
contradict the substance of the mathematician’s discipline, nor to contradict potential applications, provided of course these can be seen to be unconfused.
ii)
A crucial example is that quoted at length in App. 2, below. We might be
justified in speaking of a boundless mathematical - or linguistc - ability, to
avoid certain other confusions. What foundations
an ability so described might require is the topic of Niles, and of Nielsen.
Importantly, there surely will be (further?) disagreements here between
Wittgensteinians and (philoosphical) linguists; for the grammars of “ability”
and “capacity” and “state” etc. are liable to be murdered by the latter. E.g.
As Guetti will detail in forthcoming work, “ability” is an idealisation, of
actions actual and (mostly) possible, conceived of along static lines.
iii)
Over-indulgence in apparently-Wittgensteinian argumantation can be tantamount
to behaviourising, finitising, intuitionising, etc. . In particular, we must
not hypostatize the infinite, make it unreachable, as Sayward does, after many
‘epistemologically-oriented’ philosophers of maths have done before him. For
then we merely entrench the hopes of those we are trying to persuade to see
their ‘paradise’ for what it is, and fail to get clear on how the words
‘finite’, ‘infinite’, ‘language’, ‘ability’, etc. , are and ought practically
to be used.
“
“The objection that “the finite cannot grasp the infinite” is really directed
against a psychological act of grasping or understanding” (RFM V para.6). But
our concern here is with logical syntax: not pseudo-epistemology. Hence our
true task is to clarify the logical grammar of two completely separate
concepts: [in Waissman’s words,] ‘the crucial distinction ... between
“totality” and “system”’”.[26]
Appendix 2: The
Unstatability of Infinitude?
Some of this argument may not have been severe enough. For have we not been
trading, ourselves, on the distinction between finitude and infinitude, in
arguing that language is not infinite? This dogmatic attitude may have
committed us overly to the existence of the (potential? [here Wittgenstein’s
point that infinity is a family-resemblance concept is important]) infinite;
more importantly, it may have led us, contrary-wise to our stated intent, to partially assimilate extreme hugeness
(or endlessness to certain intents and purposes of investgiaation) to infinity. For, as pointed out in
Appendix 1, there may be no harm in calling linguistic ability ‘infinite’, if
one is able, as is so difficult, to master the point that this is not to call
such ability huge, or even very
especially remarkable, or inexplicable without a special linguistic theory.
See n.3 below; and LFM p.31, pp.140-2, and especially the following crucial
sequence from pp.255-6:
[W] “ “We aren’t talking of
anything you would call big, and
therefore not of anything infinite.”--But as long as you try to point out that
we are not treating of anything infinite, this means nothing, becuase why not
say that this is infinite? What is
important is that it is nothing big.
When
one is a child, “infinite” is explained as something huge. The difficulty is
that the picture of being huge adheres to it. But if you say that a child has
learned to multiply, so that there is an infinite number of multiplications he
can do--then you no longer have the image of something huge.
If
one were to justify a finitist position in mathematics, one should say just
that in mathematics “infinite” does not mean anything huge. To say “There’s nothing infinite” is in
a sense nonsensical and ridiculous. But it does
make sense to say we are not talking of anything huge here.
[A Student] Even when one says that a child has mastered
an infinite technique, there is even there an element of hugeness and one has
the idea of something huge.
[W] Yes, but the idea of hugeness in theat case
comes only from the word “infinite” and not from what it’s used for. By watching his work, we shouldn’t get the
idea of anything huge. The teacher does not say to himself, “Ah, fancy these
boys of ten and eleven having such vast knowledge!”
Chomsky et al precisely want the teacher so to
marvel, at our youths’ ‘cognizing capacities’.[27] But, as W.
carefully and yet ecumenically enunciates his position here, he helps undercut
the ground for their so doing. If linguistic competence is infinite, still it
is not large, and nor is it necessarily to be described and explained in a way
resembling in the slightest things that we can genuinely know more or less of.
My point (in addition) is that there may in certain circumstances be more harm in conceiving of linguistic
ability as particularly huge (if finite) if this merely fuels the demand, a priori, for a special kind of
explanation of it, and tends to snuff out certain useful attempts at gaining an
ubersicht of our grammar.
But what exactly would or could the
unstatability of infinity be?
Well,
it’s not to say there’s nothing infinite. All such dogmatism would have to be
renounced. The point is,
arguably, that any prose about infinity is either about extreme largeness; or
about not having an end (a rule, etc.); or is nonsense. The nonsense is all the
gas using the word “infinity” and its cognates and derivatives which is
self-aware enough to realize it is not merely concerned with great dimension,
but does not succeed in coherently concerning boundlessness, endlessness,
whether in a technical or a prosaic sense.
Might it not be said that the things
that most philosophers and linguists want to say ‘about infinity’ are actually
unstatable? Not, that is, to
imply that there is something true which they are trying to ‘gesture at’, or
could succeed in ‘showing’, but simply that they are speaking nonsense, that
what they think they want to say is just rubbish, unless they manage to speak
sense in one of the ways outlined above, ways which do not appear essentially to involve use of the word
“infinite” or its derivatives.
Wittgenstein
writes, crucially, “Where the nonsense starts is with our habit of thinking of
a large number as closer to infinity than a small one. // As I’ve said, the
infinite doesn’t rival the finite. The infinite is that whose essence is to
exclude nothing finite. // The word ‘nothing’ occurs in this proposition and,
once more, this should not be interpreted as the expression of an infinite
disjunction, on the contrary, ‘essentially’ and ‘nothing’ belong together. It’s
no wonder that time and again I can only explain infinity in terms of itself,
i.e., cannot explain it.” (PR
pp.157-8)
I take that last sentence to be absolutely seminal. If one
understands how the word is used , there’s no problem. But any explanation partly falsifies it, and itself (as it veers
toward (?) pure nonsense). The same is surely not so of most of language. Might
we not with profit cease to use the word “infinity” at all, given this
desperate propensity it has, and substitute the less misleading “unlimited”
(see PR p.159) or the like? And simply manipulate the transfinites, if we are
higher mathematicians. And if and when we are inclined, with those who have
been directly or indirectly criticised in this paper, to ask “How can finite
beings master infinite systems?”, might we not simply abandon the question as
less unanswerable than unstatable, unaskable, and stick instead to clarifying
our concepts, or (alternatively) refining our experimental methods, and getting
to work on uncovering the empirical
preconditions for the acquisition of language? (And was W. anxious not to say
this, only out of excessive fear of appearing
revisionistic?)
[1]
Contrast Ian Niles’s “Wittgenstein and infinite linguistic competence”,
pp.193-213 of Midwest Studies in Philosophy Vol.XVII: The Wittgenstein
Legacy (ed.s French, Uehling and Wettstein, Notre Dame, Indiana: U.Notre
Dame Press, 1992). In discussing Wittgensteinian strategems directed against
claims made by philosophers and linguists as to the foundation of
/ conditions for
infinite linguistic competence, he omits to consider that, without any
substantive prior committment to finitism, Wittgenstein (or indeed any
philosopher) might come to see, to say, that the postulated infinitary competence itself is illusory (though cf.
his note 37). I take one of the key aspects of the present paper to be the
following: that I am developing a Wittgensteinian
critique of ‘the [philosophico-linguistic] infinite’, a project I take to be, perhaps
contrary to appearances and expectations, as yet very largely unattempted
(though see n.8, below). This project, should it be successful, will knock at
least one large whole in contemporary scientistic linguistics.
[2]
Cf. Goodman’s Languages of Art (Indianapolis: Bobbs-Merrill, 1968), Part
IV. Notations require their characters and composite forms to be disjoint and differentiable, rather than dense
(as are, for instance, the irrational numbers, and, perhaps, the ‘syntax of the
visual arts’). (By understanding a
language or notation thus, I am idealising -- from the lived reality of
sentences in contexts of use, and in
relation to the language in which they have their home. For detail, see my
“What “There cannot be any such thing as meaning anything by any word” could
possibly mean”, in The New Wittgenstein, eds. Read and Crary (London:
Routledge, 2000).)
[3]
Viz. PR para.s123-4, 144.; PR p.311; LFM p.32, pp.255-6. (Of course, if my
argument below is correct, it may have to be recognised that these expressions
with which “infinite” is explained are no surer and no less metaphorical than
the word itself.
[4]
See pp.163-4f. of S.G.Shanker’s Wittgenstein and the turning point in the
philosophy of mathematics. (It is worth noting that the way I have set
things up in this paragraph resists the unhealthy tendency to speak of ‘the
infinitely large’ and ‘the infinitely small’ as ‘paradigms’ of infinity.
Infinty is a concept which, if it is a concept at all, is not commensurable
with concepts of size. There is a categorial difference, if you like, between
the vastly large or the ‘absolutely as large as possible’ on the one hand and
the infinite on the other.
[5] On this point, compare para. 45 of Part II of
Wittgenstein’s Remarks on the Foundations of Mathematics (Cambridge, MA:
MIT, 1978 (revised edition), posthumously edited and published by von Wright,
Rhees and Anscombe, transl. Anscombe), and PR para.127.
[6]
Incidentally, this is true of ungrammatical
sentences too, so long as one lays down that one cannot include just
anything (e.g. purely abstract objects) as characters of which sentences may be
composed.
[7] One might add - though this is neither
entailed by nor required by my argument up to this point - that it seems highly
likely that any actual human linguistic competence will fall well short of the
limits of the finite competence just described. For instance, is it not obvious
that one may in some sense conceive of formulable, finite sentences that would
take longer than the longest possible human attention-span to read or to parse?
[8]
For amplification, compare H.A.Nielsen’s superb piece, “How language exists: a
question to Chomsky’s theory”, Philosophical
Investigations , whose general argument is highly
consonant with mine here.
[9]
The importance of this point is hard to over-estimate. See PR para.s 124,
144-6, and (especially) 138; PR pp.306-7 and 312-3;
[10]
Should we perhaps
also QUESTION THIS ‘PICTURE’ OF THE #S GOING ON AND ON and on?; DO WE REALLY
KNOW EVEN WHAT THAT IS? ALL WE HAVE
IS THE IDEA THAT THERE IS NOTHING TO
STOP US FROM GOING ON AT ANY PARTICular point.
‘BUT AT
LEAST IT’S A RELATIVELY CLEAR ‘PICTURE’; AT LEAST, ONE THAT WE WESTERN
PHILOSOPHERS DON’T TEND TO DISAGREE OVER MUCH’... Well, Wittgenstein says: ‘the
natural numbers’ is not so much a set, still less a series, as it is the
possibility of developing endless numbers of series’(RFM pp.278-9; see also LFM p.32).
[11]
P.108 of Wittgenstein’s Lectures, Cambridge 1930-2, from the notes of John
King and Desmond Lee (ed. Lee, Oxford: Blackwell, 1980; see also LFM p.32).
I cannot here go into how precisely a line such as this lays the groundwork for
the rebuttal of the suggestion that we need to grasp an infinitary ‘grammar of
mathematics’ in order to understand
mathematics (and be mathematically ‘creative’, in the Chomskyan sense of
that word). It should be clear enough from the remarks above how I would take
this suggestion to involve a conceptual mistake (though perhaps not one of
quite the gravity of the mistake involved in claims of purely linguistic
infinitude). See also n.4, above. (Perhaps it is BEST NOT TO THINK THERE’S SUCH A THING AS
UNDERSTANDING THE INFINITE ASIDE FROM MANIPULATING IT IN MATHS. EVEN
MATHEMATICIANS DON’T UNDERSTAND IT IN ANY FURTHER SENSE, PERHAPS. ( Cf.
Wittgenstein’s vital remark: PR para.138 (p.159).)
[12]
There are other ways one could generate an apparently-endless supply of
possible-strings, such as by iterating certain verbs (e.g. ‘had’), or by
nesting quotations. These will be open to the same counter-objections I am
about to make, but furthermore are more intrinsically dubious. For instance,
the available number of nestable quotations cannot be greater than the
available number of ‘first-order’ sentences.
As for iterating verb-forms (see also below):contra some logicians, is it even clear that “Jean had had had a
good day” is a grammatical sequence? The third ‘had’, it appears to me, can
only be an idling wheel. I am fighting (in the text) on the opposition’s
strongest ground - that of the (iterable) power of ‘logical connectives’ in
natural language. (A
fuller treatment of this issue would directly address the question of
INDEFINITE VS. INFINITE, as mathematical and extra-mathematical concepts; but
this fuller treatment is not necessary to my argument here.)
[13]
“Persuasion”, on pp.123-137 of French et
al. (op.cit.) . Henceforth
PER. Cf. also pp.102-5 of RFM.
[14]
The point I am making here may be claimed to require a certain problematization
of the dualismof syntax/semantics
too, a problematization which some will find unacceptable,and for which I
cannot argue in detail at present. Briefly: of what possible
philosophico-linguistical interest could an infinite ‘purely’ syntactic capacity be to us? No more than that of a
random-noise-generating machine that we might conceive of which would go on and
on and which we could not show would ever (have to) stop producing its noises,
barring some concrete physical intervention. A putative sentence which we have
no grounds for claiming could mean something in a language-game which we can
actually imagine is not usefully described as ‘syntactically correct’. “Green
ideas sleep furiously” is no more a sentence than “Blob bling, slob, job, -
sling”. Both may have effects experientially, we may associate certain ideas
with them - but that’s it, and it’s not meaning.
Grammatico-phenomenological effects can have
nothing to do with language, conceived
of as a means of action and communication (cf. the argument of James
Guetti’s Wittgenstein and the grammar of literary experience (Georgia:
Georgia U. Press, 1993), and his “Meaningful Consequences” (jt. with myself, Philosophical Forum (Winter 1999) XXX: 4, 289-316)). The same (Guettian)
considerations will apply to the terms “infinite” and “finite”, if it is
determined that (at least outside of specific technical contexts), they are in
actuality merely devices for the production of philosophical confusion... That
is, the employment of “infinite” and related concepts in ethics, in
linguistics, and so forth, will be legible for its grammatical effects, not for
its meaningful consequences. (See the
Appendices, below.) Thus semantics cannot be irrelevant -- a purely syntactic
version of language is not language at
all, and so semantically bizarre or incomprehensible items need not be
conceded to be items of language.
[15]
In a fuller presentation, one would HAVE here
TO ADDRESS THE ISSUE OF THE VAGUE BORDERLINE BETWEEN ‘OLD’ AND ‘NEW’ vis-a-vis
rules etc.; THIS IS PART OF THE Wittgenstein’s ‘rule-following considerations’
IN GENERAL. (Indeed, ideally in the present context one would also mention and
discuss the way in which hyperbole, metaphor etc. arguably cannot be excluded
from the ‘heartland’ of lnaguage, which would complicate matters considerably.)
[16]
The argument does not exist in print -- I take it directly from Kripke’s
lectures of 1992-3. (There
is no space here to defend a point on which I agree with Kripke, and which is
pretty crucial for granting his argument salience: the move he makes from
‘phenomenology’ to LOGICAL.)
[17]
For support for this claim, see Hacker and Shanker on Wittgenstein.
[18] [[Make cohere iwht Kripke arg. :
put it in this fn?]]One might add to the counter-objections I have thus far
emphasized that there must be some upper limit to the enormity and complexity
of the strings that can be composed and yet still be decidably in the notation. What grounds do we have for saying,
say, that “x is a very large number indeed” is an English sentence, if x is an unparsably large number, and the
sentence is thus itself unparsable? If this question is unaswerable, then the
objection to my argument from mathematics would fail, because one cannot
conclude from the fact that we can legitimately speak in the abstract of a
series being infinitary that we could construct an infinite number of actual
sentences about it - even were we to live
forever! I entertain this absurd notion for a moment just to make clear
that even if one grants provisionally one form of infinitariness, others do not
necessarily follow (The same would apply
to unparsably long sentences formed by many many concatenations of shorter
sentences). The substantive point
here is closely analogous to that made by Wittgenstein in para. 38 of Part II
of the Remarks on the Foundations of Mathematics (op.cit.):
“An interesting question is: what is the connection of aleph0
with the cardinal numbers whose number it is supposed to be? Aleph0
would obviously be the predicate
“infinite series” in its application to the series of cardinal numbers and
to similar mathematical formations. Here it is important to grasp the
relationship between a series in the non-mathematical sense and one in the
mathematical sense. ...A ‘series’ in the mathematical sense is a method of
construction for series of linguistic expressions.
Thus we have a grammatical class “infinite sequence”, and
equivalent with this expression a word whose grammar has (a certain) similarity
with that of a numeral: “infinity”...
From the fact, however, that we have an employment for a
kind of numeral which, as it were, gives the number of the members of an
infinite series, it does not follow that it also makes some kind of sense to
speak of the number of the concept ‘infinite series’; that we have here some
kind of employment for something like a numeral. For there is no grammatical
technique suggesting employment of such an expression.”
We perhaps have no
good grounds, then, for supposing that
there is not an upper bound on the enormity (and on the number) of sentences that can decidably be said to be in the
language. (However, it is important
once more (as in note 7) to observe that
the argument of this note does not necessitate or necessarily eventuate from
the main lines of argument in this paper.)
[19]
[Explicating this is
perhaps crucial. Is it confused? Is it that meaningful prose is synchronicaly
finite, even though poetry might be infinite? Surely poetry too will hit
limits.]
[20]
The reader may be now be starting to worry whether the expression ‘finite
being’ is well-formed. If languages are not infinitary, still it sounds odd to
call them finitary. Perhaps these mathematical terms are just not
unmisleadingly applicable to many human phenomena? What makes us want to call
Human beings ‘finite’? I am not suggesting that humans are infinite -- I have
no idea what that means -- but I am laying some suggestive groundwork for the
Appendices below. Contra, Chomsky, McGinn, Derrida, and almost the whole
philosophical tradition, it is not clear to me that it makes sense to describe
humans as finite.
[21] I should add that a fuller
treatment of these topics from a Wittgensteinian perspective would evince the
sense in which, in ordinary discourse, ‘infinitude’ in some cases is and canwithout COMPLETE unclarity be
predicated of the non-(mathematically)infinite. Thanks to my colleague Tobyn de
Marco (especially), James Guetti, Nick Huggett, Jeff Buechner and Anne
Jacobson for discussion and
encouragement.
[22]
Let us sum up here
two of my suggestions above towards such a clear understanding:
i) If NO
INFINITE, then NO FINTITE EITHER, and vice
versa.
ii) ‘Infinity’ is perhaps always unsurveyable
except in pure maths, where it has no further consequences; and in really
ordinary speech, where it has none of the consequences that linguists and
phlosophers want to draw from it. The unsurveyable may still of course be
useful.
[23] Shanker, p.164.
[24] Ibid., p.168.
[25] Ibid., p175.
[26] Ibid., p.164.