[lit-ideas] Grice and Popper on mathematics
- From: Jlsperanza@xxxxxxx
- To: lit-ideas@xxxxxxxxxxxxx
- Date: Thu, 6 Jun 2013 20:59:35 -0400 (EDT)
Austin loved mathematics. He loved mathematics so that he cared to
translate Frege's _dull_ (in the sense of boring) book on the topic. This he
did
for Blackwell -- _the_ publishing house at Oxford -- He (Austin), being an
Oxford man, would NOT have done it for, say, Cambridge University Press. And
the Clarendon Press -- the most _prestigious_ publishing house at Oxford --
would not need it.
---
On the other hand, Grice was more interested in implicatures alla G. E.
Moore:
There are three apples in the basket.
---- Therefore, there are two apples in the basket.
This led Grice to wonder about the "Fregean" sense of "number" -- specific
number "3" -- say.
(More on this in ps).
On the other hand, Popper, typically, is interested in his own
concoctions. He divides the realm of reality in three: and thinks that Euclid's
theorem belongs in World 3 -- which is totally disrespectful, to, inter alii,
Euclid.
Popper then dismisses multiple realisability of functional states
(software) in hardware (brain), and thinks he can prove something against
'materialism' (or monism, as he prefers, since he is a triadic dualist) by
pointing
at abstract ways in which 'abstractions' (like Euclid's theorem) do not
need this or that brain realisation (in Euclid's brain, originally).
Or something like that.
In a message dated 6/3/2013 12:41:49 P.M. UTC-02,
donalmcevoyuk@xxxxxxxxxxx writes:
The problem of understanding World 3 mathematical “objects”, such as
mathematical problems and mathematical arguments and theorems, is taken by
Popper as showing that it is “generally valid” that there is “direct grasp of
World 3 objects by World 2”.
On the other hand, Grice has no preconceived agenda on this, and so, for
him, the problem of understanding world 3 mathematical objects is not
_really_ a problem.
McEvoy:
"In The Self and Its Brain, Popper seeks to illustrate this “direct”
relationship between World 3 and World 2 by discussion of a theorem of Euclid.
Popper comments (TSAIB p.548, Dialogue XI):
“Although, of course, there are some World 1 brain processes going on all
the time while World 2 is awake, and especially when it is busy in solving
problems or in formulating problems, my thesis is not only that World 2 can
grasp World 3 objects, but that it can do so directly; that is to say,
although World 1 processes may be going on (in an epiphenomenal manner) at the
same time, they do not constitute a physical or World 1 representation of
those World 3 objects which we try to grasp.
“Let me illustrate this by discussing Euclid’s theorem"
which originates, in Word 1, with EUCLID. Popper seems to be dismissive on
Euclid on this.
"Euclid" is the anglicized version of the Greek name Εὐκλείδης,
meaning "Good Glory",
fl. 300 BC.
The date and place of Euclid's birth and the date and circumstances of his
death are unknown,
There is a Statue of Euclid in the Oxford University Museum of Natural
History, that Grice would visit often.
Euclid's theorem (not called by him like this) is a fundamental statement
in number theory that asserts, rather non-intuitionistically, that there
are infinitely many (he didn't say _how_ many) prime numbers.
It is often erroneously reported that Euclid proved this result by
contradiction, beginning with the assumption that the set initially considered
contains all prime numbers, or that it contains precisely the n smallest
primes, rather than any arbitrary finite set of primes. Although the proof as
a
whole is not by contradiction, in that it does not begin by assuming that
only finitely many primes exist, there is a proof by contradiction within
it: that is the proof that none of the initially considered primes can divide
the number called q.
McEvoy continues to quote from Popper:
"that for every natural number, however large, there exists a greater one
which is a prime number; or, in other words, that there are infinitely many
primes. Certainly, Euclid had impressed upon his memory (and thus
presumably upon his brain) some facts about prime numbers, especially facts
about
their fundamental properties. But there can, I think, be little doubt about
what must have happened. What Euclid did, and what went far beyond World 1
memory recordings in the brain, was that he visualized the (potentially)
infinite sequence of natural numbers – he saw them before his mind, going on
and on; and he saw that in the sequence of natural numbers the prime
numbers get less and less frequent as we proceed. The distances between the
prime
numbers get, in general, wider and wider (although this has exceptions;
for example it seems that however far we go, there are still so-called twin
primes which are separated by just one even number; but these twin primes
get rarer too).
“Now, looking at this sequence of numbers intuitively, which is not a
memory affair, he discovered that there was a problem: the problem whether or
not the prime numbers peter out in the end – whether there is a greatest
prime number and then no further ones – or whether the prime numbers go on for
ever. And Euclid solved this problem. Neither the formulation of the
problem or the solution of the problem was based on, or could be read off
from,
encoded World 3 material. They were based directly on an intuitive grasp of
the World 3 situation: of the infinite sequence of natural numbers.”"
--- where Popper is not too clear on what he means by 'intuit' -- for
surely for any intuition, there is a correlative physical substratum behind it
(belonging to what he calls World 1).
McEvoy comments:
"We might say the problem that Euclid discovered was inherent in the
infinite sequence of natural numbers, as primes are inherent in this sequence,
and so the question of whether the primes themselves constitute an infinite
sequence is inherent in the sequence of natural numbers. But what Euclid
developed was insight, a kind of depth of understanding of this sequence,
which led him both to formulate the problem and then solve it: and Popper
stresses that the “encoded World 3 material” of a sequence of natural numbers
is not such that we can “read off” the problem or its solution. We could
write out a sequence of natural numbers say 1 to 1000 – but this “encoded
World 3 material” would not ‘encode’ Euclid’s problem in explicit World 3
terms: Euclid has discovered distinct World 3 content within the World 3
content of the sequence of natural numbers – because he has seen this
sequence contains primes, which are a distinct level of World 3 content that
is
not reducible to natural numbers, and he has seen that distinct content
raises the question whether it extends infinitely like the sequence of natural
numbers (of which it is a subset).
Popper argues that this kind of insight, or depth of understanding, cannot
be reduced to a World 1 process but should be understood as the World 2
mind working directly to grasp World 3 content. The thesis that (at least
sometimes - if not “generally”) there is a “direct” relation between World 2
and World 3 is tantamount to denying that in such cases there is any World
1 mediating between World 2 and its interaction with World 3 – so, even
where the World 3 content that World 2 grasps is related to content that is “
encoded” in World 1, what World 2 grasps may transcend the “encoded World
3 content”.
Popper wishes also to oppose the idea that some World 1 brain state must ‘
mediate’ between the World 2 mind and its grasp of a World 3 “object”".
On the other hand, Grice prefers functionalism, with multiple
realisability.
I.e. Grice would read the above to aequivocate on the word "some":
"There is some world 1 brain state".
-- as per Popper's or McEvoy's phrase -- becomes:
"There is some world 1 brain state -- OR OTHER".
I.e. Grice interprets Popper as denying that there is a SPECIFIC brain
state. For surely Popper cannot deny that there is SOME brain state.
----
Multiple realisation is a feature of the functionalism that Grice embraces.
McEvoy continues:
"and he also opposes the idea that there must be a World 1 brain correlate
for any World 2 grasp of World 3 content such that the World 1 brain
correlate is a causal determinant of the World 2 activity or is inextricably
linked by way of some one-to-one correspondence. Popper accepts that there can
be no human World 2 of mind without a World 1 brain (the destruction of a
person’s physical brain would destroy the possibility of their having World
2 activity) and that all World 2-World 1 interaction is mind-brain
interaction, but he conceives the relationship between World 1 and World 2 as
going beyond any kind of one-sided dependence of World 2 on World 1."
So perhaps the Popperian can enrich his view by reading on multiple
realisability as sustained by functionalism (NOT a type of dualism as Popper
embraces, though).
Cheers,
Speranza
http://linguistlist.org/pubs/reviews/get-review.cfm?SubID=65223
The meaning of simple numeral expressions like 'two', 'three', 'twenty-
seven' etc. has turned out to be one of the most problematic issues
within linguistic semantics and pragmatics. Part of the problem is that
there seem to be several candidates for 'the' meaning of an English
cardinal. Numerals can be used in many ways, three of which have
been the focus of discussion in the pragmatic literature of the past
thirty to thirty five years: 'two' as specifying exact cardinality, 'two'
as
specifying a lower bound and 'two' as specifying an upper bound.
Bultinck's book 'numerous meanings' is an attempt at tackling the
issue by comparing the most influential theoretical trend of the past
three decades, the so-called neo-Gricean programme, with the results
of an extensive corpus study of numerals. The book contains a
detailed discussion of the legacy of Paul Grice's theory of
conversation, with particular focus on the repercussions for the
analysis of English cardinals. It is argued that the 'conventional'
meaning of a numeral needs to be established by means of a corpus
analysis. As Bultinck subsequently aims to show, such an analysis
undermines the neo-Gricean assumption that numerals present a
lower bound in their coded meaning.
CONTENTS
Bultinck starts with an thorough discussion of Grice's original motives
and proposals (CHAPTER TWO). Crucial is the distinction between
conventional meanings and implicated meanings. Whereas the former
are to be seen as the 'coded' or 'literal' meaning of an expression, the
latter arise through inferences licensed by the assumption that the
speaker observes maxims on the quantity, quality, relevance and
manner of what (s)he says. Grice intended to keep the semantics of
expressions simple by showing that a single conventional meaning
could give rise to more than one meaning by means of conversational
implicatures. The content of the conversational principles as well as
their formalisation have subsequently been much debated and
Bultinck describes these developments in considerable detail.
While acknowledging the general success of Grice's theory and its
offspring, Bultinck argues that Grice's goal to combine a theory of
conversation with the intention of preserving the logical meaning of
logical expressions is misguided. He states that there is no
methodological justification for taking the conventional meaning of a
logical natural language expression (like 'or', 'and', 'if...then') to be
exactly that of their logical counterparts. Bultinck associates what is
conventional with what is familiar and therefore argues that frequency
data can help determine which meaning is more conventional than
others.
In CHAPTER THREE, Bultinck continues his discussion of Grice's
legacy, but now focusing entirely on the literature on numerals. Most
attention goes to the neo-Gricean line of theories that is
labeled 'minimalism' and that is inspired by Horn's 1972 notion
of 'scalar implicature', a generalisation over phenomena where a weak
item on a scale implicates the negation of the stronger items.
Minimalists argue that if the numerous meanings displayed by
numerals are to be explained by means of conversational implicatures,
then it must be the case that their coded meanings line up in an
entailment scale. So, numerals are thought to form an entailment scale
such that a sentence like ''two students came'' is entailed by the
stronger ''three students came''. By uttering ''two students came'', the
speaker therefore (potentially) implicates that the stronger alternative
is false, thus arriving at the meaning ''exactly two students came''. The
entailments are accounted for by assuming that the conventional
meaning of a numeral like 'two' is 'at least two'. In the following
chapters Bultinck aims at showing that his corpus data falsifies this
line of thinking, but in the theoretical discussion he also presents
some non-empirical counterarguments, most of which are familiar from
the literature. His most salient critique, however, is a repetition of the
methodological critique he presented in chapter two. Bultinck argues
that what Grice aimed at with his notion of conventional meaning was
a standard meaning. Bultinck proposes to identify conventional
meaning with ''familiar meaning''. Conventionality is thus equated with
a relative high level of frequency. He argues that this implies that
conventional meanings are frequent. The minimalist's choice for a
conventional 'at least' meaning, however, is not based on frequency at
all. In fact, conventional meanings are solely chosen on the basis of
their potential for conversational inferences.
In chapter three, there is furthermore a short discussion of the
underspecification account (Carston 1988), where the ''logical form'' of
a numeral is underdetermined and can be enriched by specifying
with 'at least', 'at most', 'exactly' or even 'approximately'. Some other
positions (called 'marginal' by Bultinck), like those arguing for
bilateral
conventional meanings or ambiguity, are discussed as well.
In CHAPTER FOUR, a ''general corpus analysis'' is discussed which
aims at discovering the different forms and functions of numerals. The
analysis involves one thousand occurrences of ''two'' from the British
National Corpus. In chapter five, Bultinck analyses the core meaning
of numerals, namely the cardinal one. Chapter four, however, is
focused on a more general analysis which aside from taking the
syntactic form and function into account, focuses on all possible ways
of using a numeral. Apart from the core use of the specification of
cardinality, these include the numeral as a label, the numeral as a
temporal indicator and the numeral as a mathematical primitive.
Bultinck isolates a wealth of variation in usages and discusses the
underlying corpus data in great detail. He stresses that the data
clearly demonstrate that it is a mistake to simply assume that the
meaning of numerals can be reduced to a notion of cardinality. One
clear result of the analysis, however, is a correspondence between
adnominal uses and the expression of cardinality. Almost all
adnominal numerals in some sense express the cardinality of a group.
Bultinck tries to come to a hierarchy of numeral constructions in terms
of the degree of cardinality that is involved and concludes that ''the
expression of cardinality is clearly the most important function of
'two'''
(p. 153), followed by the expression of measurement, which, as
acknowledged by Bultinck, in many respects involves cardinality as
well.
In CHAPTER FIVE, a corpus analysis is presented that focuses on
what kind of meanings cardinal uses of numerals display. It is this
analysis that is supposed to contribute to the issue of the conventional
meaning of 'two'. Again, Bultinck refers to the corpus method as ''the
methodological outcome of [Grice's] theoretical insights'' (p. 168).
Bultinck distinguishes four possible meanings (pp. 176,177):
'''at least n': necessarily n + possibly more than n;
'at most n': possibly n + not possible more than n;
'exactly n': necessarily n + not possible more than n + not possible
less than n''; and
'''absolute value n': non-modal, the group of elements denoted by the
NP is determined as having n elements''
Crucial here is the assumption that the first three of these meanings
involve modal statements about cardinality. The 'absolute value n'
meaning, on the other hand, is relatively simple. In fact, Bultinck
maintains that it is 'cognitively' simple, since it refers to nothing more
than cardinality of a group, and that the other interpretations are
therefore in some sense marked. That is, the first three meanings
make what is said (understood in a non-Gricean way) about the
cardinality much more prominent than the 'absolute value'
interpretation does.
The majority of occurrences of 'two' turn out to be either of
the 'absolute value'-type or of the 'exactly n'-type. Bultinck notices
that
the 'exactly n' readings are mostly caused by definite markers. There
are no findings in the corpus of 'absolute value' uses with such
markers. This observation also serves to explain the distribution of the
different usages over different syntactic positions. For instance, the
majority of direct objects contain numerals of the 'absolute value' type,
whereas the majority of numerals in adverbial phrases are used
as 'exactly n'. According to Bultinck this distribution is simply a reflex
of
the attested fact that direct objects are generally good candidates for
introducing new topics, whereas it is less likely that material in
adverbial phrases is there to (existentially) introduce a new referent.
In subject position, occurrences of 'two' without definite markers are
mostly 'absolute value' or 'exactly n'. But the difference between these
two usages is blurred. The trend is that subject position indefinite
numerals are less likely to allow for a subsequent revision of the
involved cardinality than object indefinite numerals. Bultinck proposes
that this is due to the fact that it is marked to use a subject for the
introducing of a new referent. The focused use of the numeral hints at
excluding the possibility of there being more than the 'n' elements that
are expressed. This means that there is a continuum from 'absolute
value' to 'exactly n' meanings. In 'pure absolute value' use there is a
neutrality toward the possibility of there being more elements. This
neutrality is reduced in subject position. A further finding supports this
idea of a continuum. In predicative constructions (such as existential
there sentences), most samples show the absolute value meaning of
the numeral. Bultinck's idea is that such constructions hardly change
the default 'absolute value' interpretation of the numeral. Although
Bultinck is careful not to present it as a clear result from his corpus
research, he hypothesises that the continuum from 'absolute value'
to 'exactly n' is paired with a scale of syntactic constructions, ranging
from existential there sentences, to objects, to subjects, to adverbials.
The picture emerging from this is one where a great multitude of
factors influence the 'value interpretation' of a numeral. In particular,
it
seems generally the case that when there is an 'exactly n'
interpretation of the ''meaning complex'' that contains the numeral, this
meaning can be reduced to a combination of the 'absolute value'
meaning and the influence of other co-textual factors. It follows
that ''[the] 'absolute value' interpretation is the starting-point for the
interpretation of 'two''' (p. 225), or as Bultinck concludes in chapter
six, ''the conventional meaning (the ''coded content'') of 'two' is
the 'absolute value' meaning'' (p. 307).
The corpus analysis shows that 'at least n' uses of numerals are
highly infrequent (3,9%). This, Bultinck claims, is highly problematic for
the neo-Griceans. In fact, the corpus analysis shows that the few 'at
least' uses that are found are all due to the co-occurrence with a
linguistic element and, in most cases, that element is 'at least'.
Another finding from the corpus discredits the neo-Gricean account of
numerals in another way. One of the traditional arguments for
assuming the 'at least n' meaning to be conventional is that
were 'exactly n' conventional, then it would be redundant to combine
the numeral with 'exactly'. It is not and hence, the argument
goes, 'exactly n' cannot be the coded meaning of 'n'. The corpus
shows, however, some very clear facts about numeral modifiers
(called 'restrictions' by Bultinck). The most common kind of
modification is with 'at least' (44.8%), whereas combinations of 'two'
with 'exactly' are relatively rare at 9.5%. If the neo-Gricean argument
holds, exactly the reverse distribution of 'exactly' and 'at least' would
be expected.
CHAPTER SIX briefly sums up the results of Bultinck's work and
repeats the general conclusions.
CRITICAL EVALUATION
The first half of the book is devoted to the discussion of the literature
on (neo-)Gricean implicatures in general and the pragmatics of
numerals in particular. A shorter discussion might have been more
effective, since one has to wait a long time for Bultinck's main feat, the
discussion of his corpus study of numerals (chapters four and five).
Furthermore, the literature discussion is often overly detailed and
repetitive. For instance, some of the arguments Bultinck discusses in
the chapter on Gricean pragmatics are repeated in both his discussion
of the literature on numerals and in the discussion of the corpus data.
Nevertheless, it is certainly admirable that Bultinck so successfully
weaves together discussions from linguistic pragmatics, corpus
linguistics and cognitive linguistics. Although tedious at some points,
the many repetitions might actually guarantee that this book is suitable
for the broad audience it sets out to reach.
A more serious problem is the fact that the discussion in chapters two
and three is in many ways dated. Browsing the references, one finds
that the most recent literature that is being discussed dates from 2001
(the book is published in 2005). Of course, many of the high points of
the discussion of scalar implicatures can be traced back to the 1970s
and 1980s, so it is perhaps not entirely unexpected to find mostly
older literature. However, in the past few years the study of
implicatures and numerals has flourished once again. Now, there is a
wealth of new findings and theoretic proposals (e.g. Geurts 1998,
Chierchia 2002, Recanati 2003, van Rooy and Schulz 2004).
Furthermore, an increase in the interest of psycholinguists into
pragmatic issues has lead to a considerable amount of empirical data
challenging the traditional theoretic approaches to make more precise
predictions (see, for instance, Noveck 2001, Papafragou and
Musolino 2003 and, especially, Musolino 2004). Unfortunately, such
recent works are completely absent from Bultinck's discussions and
arguments. This may be explained by the fact that this book, as I
understand it, is a published version of Bultinck's dissertation which
dates from 2001. Curiously, however, this fact is not mentioned in the
book.
The main objective of Bultinck's corpus analysis seems to be to
discredit the idea that numerals carry a conventional meaning that
involves a lower bound. With this in mind, I think the three most
relevant findings are: (A) the corpus is argued to display the
infrequency of this alleged coded meaning; (B) the data suggest that
there are 'numerous meanings' associated with English cardinals and
that these are less rigidly distributed than the neo-Gricean programme
would have it; and (C) the 'absolute value' meaning is the most basic
one of these numerous meanings.
It is not entirely clear to what extend Bultinck's 'at least n' meaning
corresponds to the lower bound conventional meaning defended by
the minimalists. I doubt whether the neo-Griceans really had a modal
coded meaning in mind. It is certainly not the case that the lower
bound meaning necessarily involves modality. It is quite easy to
imagine a 'cognitively simple' lower bound analysis which simply
describes the cardinality of a group as being 'greater or equal than n'.
In fact, such a proposal comes very close to Bultinck's own 'absolute
value' meaning. This becomes clear from Bultinck's specification of the
four candidate meanings. The 'at least n' meaning is described
as ''necessarily n + possibly more then n'' (p. 176). Note that in this
definition, one needs to assume that the number symbol 'n' has a
greater-or-equal reading itself. If the cardinality of a group is
necessarily 'n', how can it at the same time be possible that this
cardinality is 'more than n'? A formulation like this one presupposes
once again that numerals somehow line up in entailment scales. It
follows that the 'absolute value' meaning is really a lower bound
meaning. Consequently, one could characterise Bultinck's proposal as
minimalistic, except that the conversational implicatures have been
replaced by co-textual factors that trigger modal cardinality statements.
So how well does this proposal account for the data? The 'absolute
value' meaning of numerals seems consistent with the data in the
corpus. It is important, however, to explain in detail how the
compositional meaning of numerals is defined, especially since these
very meanings have turned out to be so remarkably deceptive.
Unfortunately, the semantic processes Bultinck refers to are often not
specified enough to assess how the sentential meanings are derived
from a single core lexical meaning.
Nevertheless, 'numerous meanings' contains a wealth of data and
ideas that will stimulate the ongoing discussion of the semantics of
simplex and complex English numerals. Anyone working on a linguistic
topic that is somehow related to numeral meaning will definitely find a
lot to learn in this book, especially since Bultinck's most important
point, I feel, is not theoretical but methodological. The data are much
more varied and complex than the neo-Gricean theories have
assumed. On the basis of this, Bultinck argues convincingly that it is a
mistake to search for 'the' meaning of English cardinals.
REFERENCES
Carston, R. 1988. Implicature, explicature and truth-theoretic
semantics. In Kempson, R. (ed.), Mental Representations: The
interface between Language and Reality.
Chierchia, G. 2004. Scalar Implicatures, Polarity Phenomena, and the
Syntax/Pragmatics Interface. In Belletti, B. (ed.), Structures and
Beyond: The Cartography of Syntactic Structures. Vol. 3. New York,
NY: Oxford University Press.
Geurts, B. 1998. Scalars. In Ludewig, P. and Geurts, B. (eds.)
Lexicalische Semantik aus Cognitiver Sicht. Tuebingen: Gunter Narr.
95-117.
Horn, L. 1972. On the Semantic Properties of Logical Operators in
English. UCLA dissertation. Distributed by Indiana University
Linguistics Club, 1976.
Musolino, J. 2004. The semantics and acquisition of number words:
Integrating linguistic and developmental perspectives. Cognition 93(1):
1-41.
Noveck, I. 2001. When children are more logical than adults:
Experimental investigations of scalar implicature. Cognition 79: 165-
188.
Papafragou, A. and Musolino, J. 2003. Scalar implicatures:
Experiments at the semantics-pragmatics interface. Cognition 86(3):
253-282.
Recanati, F. 2003. Embedded Implicatures, Philosophical Perspectives
17(1): 299-332.
van Rooy, R. and Schulz, K. 2004. Exhaustive interpretation of
complex sentences. Journal of Logic, Language and Information, 13:
491-519.
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