A Practice-Based Approach to Diagrams by Valeria Giardino

Abstract In this article, I propose an operational framework for diagrams. According
to this framework, diagrams do not work like sentences, because we do not apply a set
of explicit and linguistic rules in order to use them. Rather, we become able to manipulate
diagrams in meaningful ways once we are familiar with some specific practice, and
therefore we engage ourselves in a form of reasoning that is stable because it is shared.
This reasoning constitutes at the same time discovery and justification for this discovery.
I will make three claims, based on the consideration of diagrams in the practice of logic
and mathematics. First, I will claim that diagrams are tools, following some of Peirce’s
suggestions. Secondly, I will give reasons to drop a sharp distinction between vision and
language and consider by contrast how the two are integrated in a specific manipulation
practice, by means of a kind of manipulative imagination. Thirdly, I will defend the idea
that an inherent feature of diagrams, given by their nature as images, is their ambiguity:
when diagrams are ‘tamed’ by the reference to some system of explicit rules that fix their
meaning and make their message univocal, they end up in being less powerful. p. 135

Rigor might preserve truth, but insight, understanding, and
explanation hint at reasons. Furthermore, there are proofs that explain—which might be
referred to as causal proofs—and proofs that convince but do not explain—which can be referred to as non-causal proofs [3]. If we look at the practice of mathematics, we
realize that verification is proof, but verification might not provide reasons: mathematicians
are not satisfied with proving conjectures, since what they want is reasons for these
conjectures [23].

Views such as the suspicious view, focusing on rigorous formal proofs, move away
from the consideration of actual mathematical practice, both in contemporary mathematics
as well as in the history of mathematics. According to Corfield, they apply what he
calls the ‘foundational filter’; because of this filter the only interesting questions in philosophy
of mathematics are about the possible reduction of mathematics to some foundational
system [7]. p. 137-8

Nevertheless, the assumption behind the claim that diagrams are never sufficiently
general and never sufficiently appropriate is that diagrams are depictions—though partial
and imprecise—of abstract objects. I want to contend this claim and propose that diagrams
are not pictures of abstract objects but tools for reasoning about abstract relationships. p. 139

First, in order to be effective, diagrams must always be interpreted within a certain
context of use. Secondly, the reference to them contributes to the very definition of this
context and gives structure to the problem to solve. Thirdly, diagrams belong to the genus
of ‘representation’ as “that character of a thing by virtue of which, for the predication of a
certain mental effect, it may stand in place of another thing” ([13], vol. 1, par. 564; written
in 1893). Nevertheless, this should not be taken literally, for diagrams do not ‘directly’
depict some abstract object whose existence is presupposed; rather, they embody a selection
of relevant relations. Finally, diagrams are given with an intention, as all tools are,
cognitive and epistemic tools being among them: they are conceived so as to achieve some
particular aim, and the intention behind their creation must be acknowledged in order to
appropriately interpret and use them. p. 141

An important advantage of this operational approach is that it discards the opposition
between visual reasoning and linguistic knowledge. In fact, the dichotomy of vision vs.
language, which has led to the antithesis between visuocentric and logocentric views, is
pernicious. p. 141

Diagrams are always considered from within a specific practice and context. Imagine
the diagram of a circle. It has been used in logic, for example, by Euler, in Euclidean
geometry, and in Cartesian geometry.Was it ‘the same’ circle in all these cases? Did it play
the same role? Of course it did not. In fact, there is nothing like a specific set of rules that
could be fixed once for all for all circles or for some specific kind of diagram in general.
My proposal is that in diagrammatic reasoning what counts is not the appearance of a
diagram and a list of explicit rules that can be applied to it, but rather a set of procedures: when one learns to use a certain diagrammatic system for performing some inferences,
she learns a manipulation practice. The diagram becomes the mathematician’s worksite,
where operations, plans, and experiments are made in order to find solutions and reasons
for these solutions. While syntactic rules are piecemeal, procedures are holistic. p. 145-146

My hypothesis is that, inside a specific practice, the space of the diagram perceived
combines with the actions actually performed or imagined on it, in continuous interaction
with linguistic knowledge. All these elements together contribute to mathematical
meaning-making: manipulative imagination is at work to provide evidence in favor of
some particular train of thought. p. 149

The dichotomy between visual thinking on the one hand and linguistic processes on
the other has obscured the fact that what counts in diagrammatic reasoning is the manipulation
practice, based on holistic procedures and not on the definition of explicit linguistic
rules. According to this practice-based framework, it is the practice that fixes the meaning
of the diagrams on each occasion, otherwise they are as ambiguous as other images are,
and it is a kind of manipulative imagination that operates on them. Each practice is defined
by procedures of manipulations and interconnected facts. All these elements taken
together define in turn the system of knowledge shared by the community; this system
encompasses diagrams, statements, particular notations and actions prescribed or allowed
on them. p. 150



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