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. 45

b. 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. 47

c. 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. 48

I 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. 51-2

It 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. 52

Going 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. 59

But 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



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