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