Introduction: From Practice to Results in Mathematics and Logic by Valeria Giardino, Amirouche Moktefi, Sandra Mols et Jean Paul Van Bendegem

This looks to be from a special volume of Philosophia Scientiæ, 16 (1), 2012, 5–11.

This volume is a collection of essays that discuss the relationships between the practices deployed by logicians and mathematicians, either as individuals or as members of research communities, and the results from their research. We are interested in exploring the concept of ‘practices’ in the formal sciences. Though common in the history, philosophy and sociology of science, this concept has surprisingly thus far been little reflected upon in logic and mathematics. Yet, such practice-based approach would be most crucial for a critical study of these fields. Indeed, in their daily work, mathematicians and logicians do deploy sets of practices, some intimately tied to mathematics and logic as such and others relating rather to material, theoretical, intellectual and social issues in their environments.  For these reasons, this volume pertains to the interdisciplinary trend that has gained more and more interest in recent years, and which is known as the philosophy of mathematical practice. p. 5-6

There are a lot of good references in here:

To give an all too short account of this recent approach, one might look
for its roots in Imre Lakatos’s 1976 seminal Proofs and Refutations [Lakatos
1976], a work that was in part inspired by Georg Polya’s How to Solve It
[Polya 1945] which discusses heuristics and problem-solving techniques in an
educational setting. Later on, Philip Kitcher proposed in his book The Nature
of Mathematical Knowledge [Kitcher 1985] a more or less formal model of
how mathematics as an activity can be described.

[…]

Independently of these developments
in the philosophy of mathematics, but also partially inspired by
Lakatos as well as Thomas Kuhn’s The Structure of Scientific Revolutions
[Kuhn 1962], some researchers developed a sociology of mathematics, where
one of the major focuses was mathematical practice as a group or community
phenomenon. Two works should be mentioned in this line: David Bloor’s
Knowledge and Social Imagery [Bloor 1976] and Sal Restivo’s The Social
Relations of Physics, Mysticism, and Mathematics [Restivo 1985]. p. 6

Looks like I’m interested in the micro level:

An elegant way to obtain some coherence in such an agenda
is to look at the various levels where the practices are situated: macro-, mesoand
micro-level, as a first rough classification. p. 7

At the micro-level, the core theme is the nature of mathematical proof.
Besides the notion of proof itself and its inner difficulties, there is a wealth of related
themes to explore. A first group includes issues related to presentation—
concerning the (non-)formalized character of a proof, its (in-)formality, the use
of diagrams and their status —, and accessibility—who is able to grasp the
proof?—and, perhaps most importantly, why is a proof convincing? A second
group of themes concerns what ‘accompanies’ proofs. Mathematicians
perform ‘experiments’, but are such quotation marks necessary? Why (not)?;
they crunch numbers in the hope of finding interesting patterns, performing
inductive reasoning; they support their proofs by arguments, leading to the
questions whether proofs themselves can be seen as a particular type of argument,
and whether, fascinatingly, a rhetorics of mathematical texts is possible.
Quite interestingly, at this latter micro-level, other fields and disciplines than
history, philosophy and sociology, also contribute to the picture. Such is the
case with cognitive psychology, crucial to understand reasoning processes as
they take place in the human (social?) brain. Such is the case too with evolutionary
biology and psychology, necessary to establish the roots and origins of
mankind’s ‘number sense’, if any. It is to be noted too that, at this micro-level,
the impact of the particular tradition, whether ‘maverick’ or ‘naturalized’, is
not overwhelmingly present, making a close collaboration perfectly possible. p. 8

The essays are summarised as follows:

In the first group of essays, first, as a general discussion, Mark Smith defends
a conception of mathematical practice and mathematical subject matter
that puts together inferential pluralism and a form of concept-realism. The
second essay by Danielle Macbeth is a close examination of the relation between
the practice of proving and understanding. Fully formalized mathematical
proofs usually do not advance our mathematical understanding: does this
mean that form does not correspond to content? In her view, to avoid such
an opposition, it is necessary to develop a different notion of (formal) proof,
and eighteenth-century algebraic proofs can provide an example of fully rigorous
and fully content-filled mathematical proofs. The third essay, by Jessica
Carter, is an analysis of the role of representations in mathematical proofs,
illustrated by an example from contemporary mathematical practice where
the value of an expression is found by gradually breaking it down into simpler
parts. More generally, and referring to the Peircean terminology, she also discusses
the role of icons and indices in such a practice. Finally, the fourth essay
by Catarina Dutilh Novaes delineates a practice-based philosophy of logic and
illustrates it by focusing on the role played by formal languages in logic. In her
view, formal languages have an operative role; paper-and-pencil and hands-on
technology trigger specific cognitive processes, which psychology discloses.
In the second group of essays, the first three are analyses of case studies
exploring specific mathematical practices. First, Daniele Molinini focuses on
the explanatory character of Euler’s proof of his Theorem, which, according to
the author, is not recognized by Steiner’s approach to mathematical explanation.
Secondly, Irina Starikova shows how the representation of groups through
graphs leads to a new conceptual perspective on the geometry of groups.
Thirdly, Baptiste Mélès defends the thesis that the concepts of paradigm and
thematization used by Jean Cavaillès find an illustration and a formalization
in category theory and a precedent in the Hegelian dialectic. In the fourth and
last essay, Vincent Ardourel explores possible interactions between mathematics
and physics, by focusing on instances of practices of mathematical research
that consist in reformulating constructively physical theories. p. 10

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