Exercise books. Handmade paper and pencil.

# Grids

Taking grid-drawing to its limit – drawing out the guidelines again and again to make the surface to work from.

# Relevance and rationality by NICHOLAS ALLOTT

5.2 Optimization under constraints

We have established that no realistic model of reasoning, including therefore the

relevance theoretic comprehension procedure, can be an example of unbounded

rationality. Next I will consider optimization under constraints. This vision of

rationality ‘holds that the mind should calculate the benefits and costs of searching

for each further piece of information and stop search as soon as the costs outweigh

the benefits.’ (Gigerenzer and Todd, 1999, section 2.2)

Therefore a solution is reached without consulting all of the evidence, in contrast

to the way models of unbounded rationality work, using all possible information.

However the requirement that at each stage the costs and benefits of containing the

search be calculated leads to a computational explosion: ‘the paradoxical approach

is to model “limited” search by assuming that the mind has essentially unlimited

time and knowledge with which to evaluate the costs and benefits of future

information search.’ (Gigerenzer and Todd, 1999, section 2.2) This means that

‘optimization under constraints can require even more knowledge and computation

than unbounded rationality.’ (op cit, section 2.2, referring to work by Vriend,

1996; Winter 1975)

As a realistic model of cognition, then, relevance theory cannot rely on

optimization under constraints; indeed it does not, but there are two reasons why

someone might suppose that it does. First, according to the communicative

principle of relevance, ‘Every ostensive stimulus conveys a presumption of its own

optimal relevance’ (Sperber and Wilson 1986/95, p 158). This means that the

hearer is licensed to search for an optimally relevant interpretation: ‘An ostensive

stimulus is optimally relevant to an audience [if] it is the most relevant one

compatible with the communicator’s abilities and preference.’ (Wilson and

Sperber, 2002, section 3) So the relevance theoretic comprehension procedure

looks for the optimal solution, given a particular stimulus in a particular context.

Thus it appears to be an optimizing procedure, but as I shall argue, it is not a kind

of optimization under constraints.

Secondly, as previously noted, relevance is a matter of effort and effects, so the

generalization ‘stop when your expectations of [optimal] relevance are satisfied’

(Wilson and Sperber, 2002, section 3) may seem to be an injunction to calculate at

each stage the costs and benefits of continuing with the search and to stop when

the projected costs in effort outweigh the prospective benefits in cognitive effects.

This is a misinterpretation, however, for two reasons. First, the hearer is licensed

to ‘stop at the first interpretation that satisfies his expectations of relevance,

because there should never be more then one.’ (Wilson and Sperber, 2002, section

3) This follows from the special nature of ostensive-inferential communication: the

speaker ‘wants her utterance to be as easy as possible to understand so that the first

interpretation to satisfy the hearer’s expectations of relevance is the one she

intended to convey.’ (op cit. , section 3) This means that there is no need to

calculate the costs and benefits of continuing the search: what is at issue is rather

whether the cognitive effects are (more than) enough at some time, t, to justify the

processing effort incurred from the beginning of the search to that time.

The second reason why the relevance theoretic comprehension procedure could

not be a species of optimization under constraints is that from the beginning

Sperber and Wilson have been clear that ‘contextual effects and processing effort

are non-representational dimensions of mental processes’ (1986/95, pp 131). We

may sometimes have intuitions about degrees of effort and effect but efforts and

effects – and therefore relevance – are not generally mentally represented and

therefore cannot be used in computations. Thus there is no possibility that future

effort and effects could in general be summed and weighed up against each other

as optimization under constraints requires. p. 77-78The argument that the relevance theoretic comprehension procedure is a type of

satisficing procedure comes from comparing Gigerenzer’s definition, ‘satisficing

(sets) an aspiration level and ends the search for alternatives as soon as one is

found that exceeds the aspiration level’ (Gigerenzer and Todd, 1999, section 2.3,

referring to work by Simon, 1956, 1990) with the specification of the relevance

theory comprehension procedure:

(a) Follow a path of least effort…(and)

(b) Stop when your expectations of relevance are satisfied. (Wilson and

Sperber, 2002 section 3)…

In the case of ostensive-inferential communication,

‘relevance theory claims that use of an ostensive stimulus may create precise and

predictable expectations of relevance not raised by other stimuli.’ (Wilson and

Sperber, 2002, section 3) This is because ‘an ostensive stimulus is designed to

attract the audience’s attention. Given the universal tendency to maximise

relevance an audience will only pay attention to a stimulus that seems relevant

enough.p. 78-79

# History and Intentions in the Experience of Artworks by Alessandro Pignocchi

This paper sketches a model of the

14 experience of artworks based on the mechanisms of

15 intention attribution, and shows how this model makes it

16 possible to address the issue of personal background

17 knowledge empirically. I claim that the role of intention

18 attribution in art experience has been incorrectly accounted

19 in the literature because of an overly narrow definition of

20 ‘‘intention.’’ I suggest that the observer can recover not

21 only the artist’s abstract projects, but any kind of mental

22 states that have played a causal role during the production

23 of the work. In addition, I suggest that this recovery occurs

24 in large part unconsciously and/or implicitly. p. 1researchers in the 54

humanities doubt the existence of artistic universals, 55

arguing that the way we evaluate an artwork always 56

depends on what we know about its context of production 57

(Danto 1981). For instance, we will not attribute the same 58

value to an impressionist painting if we believe that it was 59

painted in 1872 or last year (Genette 1997). p. 1106 Although some authors have denied any role for intention

107 attribution in the experience of artworks (Wimsatt and

108 Beardsley 1954), nowadays the majority recognizes that

109 intention attribution must play some role (Danto 1981;

110 Iseminger 1992; Levinson 1979; Walton 1970). However,

111 this role may have been incorrectly described, or, at the

112 least, some of its important components may have been

113 neglected (Pignocchi 2012). This is due, first, to an iden-

114 tification of the artist’s intention with her conscious and

115 abstract aims, as if a work of art could be produced on the

116 basis of a single intention or a small set of them. p. 2Discussing Painting as an art,

179 (Carroll 2011) claimed that Wollheim position should be

180 called ‘‘mentalist’’ and not ‘‘intentionalist’’ since it con-

181 siders many kinds of mental states, including unconscious

182 one, that are not traditionally considered as intentions.

183 However, I want to try to subsume all of these mental states

184 under the common label ‘‘intention,’’ to insist on the notion

185 of causality. p. 3The take-home message of this article is that the 874

intention attribution that determines part or all our expe- 875

rience of an artwork and of its properties can be implicit 876

and unconscious. Thus, we frequently do not notice their 877

influence. We have the impression of liking or disliking the 878

work or its properties per se. We say that we ‘‘like’’ this 879

artwork, without noticing that what we actually like is the 880

content of the intentions that we see behind it (we find them 881

sincere, sophisticated, original, coherent, audacious) and 882

their skilful realization in the work. p. 9

# Linguistic anthropology: Language as a non-neutral medium by Alessandro Duranti

[To appear in The Cambridge Handbook of Sociolinguistics, Edited by Raj Mesthrie]

In this chapter I focus on

three essential properties of language that are usually assumed by linguistic

anthropologists: (1) language is a code for representing experience, (2) language

is a form of social organization, and (3) language is a system of differentiation. p. 4The idea that in using a given language speakers are forced into

interpretations of the world that they cannot quite control dates at least as far

back as the writings of Johann Gottfried Herder and the diplomat and linguist

Wilhelm von Humboldt (see Bauman and Briggs 2003: Chapter 5). p. 5Implicit in this line of work is that the notion of habitus has become

associated with a conceptualization of language as a practice that is quite

different from the ways in which language has been conceived of in the literature

on linguistic relativity as discussed above. In this new perspective, which

characterizes what I have elsewhere called the “third paradigm” in linguistic

anthropology, language is viewed as being composed of more than just lexicon

and grammar. It also includes communicative resources such as prosody, tempo,

volume, gestures, body posture, writing tools and conventions, and visualization

(see for example Goodwin 2000; Finnegan 2002). p. 13After the publications of two

posthumous works of two philosophers – Ludwig Wittgenstein’s (1953)

unfinished Philosophical Investigations and J.L. Austin’s (1975) lectures How To Do

Things With Words –, an increasing number of scholars began to see language

predominantly as action rather than mostly (or exclusively) as a code to express

ideas or represent events. p. 16-7Building on these insights and in interaction with a number of innovative

scholars (e.g. Kenneth Burke, Erving Goffman, John Gumperz, William Labov),

starting in the mid-1960s Dell Hymes began to alter the object of study of earlier

generations of linguistic anthropologists by shifting the attention from ‘language’

(a system, e.g. a grammar) to ‘speaking’ (an activity, e.g. telling a story). p. 18In the 1960s no one could have agreed more with the idea that language is a

form of social organization than a group of sociologists who became known as

“conversation analysts.” This explains the inclusion of articles by Harvey Sacks

and Emanuel Schegloff in Gumperz and Hymes’ (1972) edited volume Directions

in Sociolinguistics: The Ethnography of Communication. Sacks and Schegloff were

arguing within sociology that one should study conversation as a prominent site

of social organization. p. 19Research on language ideology is closely related to but still distinct from

Pierre Bourdieu’s (1991) concept of symbolic domination. In his view, the social

value of the language varieties that we speak (e.g. the dialect or dialects we are

comfortable with, the register range) is given by the place of such varieties within

a linguistic market that the individual cannot control. Therefore, for Bourdieu, as

users of particular linguistic language varieties we are the victims of a system of

social discrimination that has profound consequences for our chances to succeed

in society. p. 27

# The Cognitive Basis of Mathematical Knowledge by Marinella Cappelleti and Valeria Giardino

This paper surveys a lot of interesting stuff about cognition and mathematics along the Dehaene lines. Interestingly, they distinguish the cardinal, ordinal and nominal functions of numbers and give evidence that these come apart.

# The Role of Notation in Mathematics by Edwin Coleman

This is a thesis, of which I only have a very short summary. One point useful to me is that he describes mathematical writing as divided up into words, diagrams, notation and pragraphy.

# Mathematical Reasoning and External Symbolic Systems by Catarina Dutilh Novaes

This paper is nice in that it discusses several cultures and draws upon Stanislas Dehaene and similarly experiment-based data.

It is an almost trivial observation that the practice of mathematics typically

involves a lot of ‘scribbling and fiddling’ with symbols, diagrams and special

notations. Taking as a starting point the idea that both the written and the

oral languages used by mathematicians are philosophically relevant aspects

of their practices, the aim of this paper is to discuss in more detail the exact

status of external symbolic systems, systems of writing in particular, for

mathematical reasoning and mathematical practice. Are they merely convenient

devices? Are they essentially heuristic components? Can mathematics

be practiced without recourse to symbolic systems? In what sense, if any,

can different forms of writing be said to be constitutive of doing mathematics? …Indeed, the investigation takes

into account three different levels: the synchronic level of a person ‘doing

math’ at a given point in time; the diachronic, developmental level of how an

individual learns mathematics; and the diachronic, historical level of the development

of mathematics as a discipline through time. It will be argued that

the use of external symbolic systems is constitutive of mathematical reasoning

and mathematical practice in a fairly strong sense of ‘constitutive’, but

not in the sense that manipulating notations is the only route to mathematical

insight. p. 45b. Mathematical reasoning is conducted in vernacular languages; notations

are merely convenient short-hands

This position, as described by Macbeth, views mathematical reasoning as

constitutively independent of special systems of notation, but as inherently

tied to vernacular languages. p. 47c. Mathematical reasoning is not language-dependent

On this view, to ‘do’ mathematics would be a purely private, inner process,

which can then be expressed and communicated a posteriori in some public

medium such as systems of notations or spoken languages.Thus, positions b. and c. both reject the notion that mathematical (written)

notations are constitutive of mathematical practice. Positions a. and b. have

in common the idea that mathematical reasoning requires some sort of external,

linguistic medium to come about (as opposed to thoroughly internalist

position c.), yet disagreeing on the exact nature of this medium. p. 48I here argue that, based on empirical data drawn from different

fields, a strong case can be made for the claim that external media are

constitutive of mathematical knowledge and mathematical reasoning in the

stronger sense that, even when a given person is apparently not manipulating

symbols, such as a mental calculator, she in fact typically relies extensively

on internalized versions of external devices (at least in most cases). p. 51-2It is known for

example that the fastest and least error-prone mental calculations are those

consisting in adding a given number smaller than 10 to a multiple of 10 (10,

20, 30 etc.); this is arguably because the reasoner mentally ‘replaces’ the 0

on the right-side of one of the numerals with the other numeral.9 So it seems

that, while not using ‘paper and pencil’ at that particular moment, the operation

being implemented relies significantly on the mode of presentation of

the Hindu-Arabic numeral system as a place-value system, and thus on an

internalization of external symbols. p. 52Going back to the three positions presented above, it seems clear that, even

if at specific occasions (i.e. the synchronic level), ‘doing math’ does not require

the act of manipulating external symbols (as the defender of position

c., the ‘documentist’, would have it), from a diachronic, developmental point

of view, external symbols appear to be a necessary condition for the emergence

of mathematical concepts and mathematical reasoning. Moreover, I

have argued that many of the processes which appear to take place exclusively

‘in the head’ are in fact internal simulations of external processes; in

such cases, there is a clear sense in which external representations are constitutive

of mathematical reasoning, even if they are simulated and manipulated

mentally. p. 55

She takes two interesting cases, that of a savant and a blind mathematician, as interesting exceptions to prove the rule

Many mathematicians see numbers as digits, and while they can do

amazing things when it comes to understanding the very root of an

equation, only when you ‘see’ it and do the mathematics together

can you really understand where it comes from.

Padgett’s claim that many (most?) mathematicians ‘see numbers as digits’

is indeed very much in the spirit of the views defended here. But his final

observation is what is most revealing: he refers to a form of ‘seeing’

the root of an equation that is independent of seeing numbers as digits, and

thus a form of mathematical insight which is presumably not inherently tied

to external representations (crucially, the fractals he draws are renditions of

what he ‘sees’ prior to making the drawings themselves). Notice however

Padgett’s suggestion that both this ‘seeing’ ability and ‘doing the mathematics’

(presumably, operating with symbols) are required to understand where

the root of an equation ‘comes from’. p. 59But rather than disproving the claim that mathematical

practice is fundamentally tied to forms of writing, at least some blind

mathematicians seem in fact to confirm it in that the ways in which they produce

mathematical knowledge are often significantly different from those of

sighted mathematicians. To use Jason Padgett’s terminology, we might say

that, unlike most mathematicians, blind mathematicians arguably do not predominantly

see ‘numbers as digits’, and at times this seems to provide them

with privileged insight with respect to some specific problems. p. 61