Proposed by:

Jessica Carter (Univ. of Southern Denmark, Odense, Denmark)
Marco Panza (IHPST Paris, France)

Participants:

Andrew Aberdein 
Karine Chemla 
Albrecht Heeffer
Bart van Kerkhove 
Ladislav Kvasz 
Javier Legris 
Jemma Lorenat 
Benedikt Löwe 
Danielle Macbeth 
Madeline M. Muntersbjorn
Philippe Nabonnand
Jairo da Silva 
James Tappenden

Rationale:

The philosophy of mathematics has experienced a very significant resurgence of activity
during the last 20 years, much of it falling under the widely used label “philosophy of
mathematical practice”. As a reflection of this state of affairs, in 2009 a group of nine
researchers in this field gathered to promote the creation of the Association for the
Philosophy of Mathematical Practice, APMP.

Approaches to the philosophy of mathematics that focus on mathematical practice have
been thriving. They include the study of a wide variety of issues concerned with the way
mathematics is done, evaluated, and applied, and in addition, or in connection therewith,
with historical episodes or traditions, applications, educational problems, cognitive
questions, etc. We use the label “philosophy of mathematical practice” as a general term
for this gamut of approaches, clearly open to interdisciplinary work. APMP members
promote a broad, outward‐looking approach to the philosophy of mathematics which
engages, with mathematics in practice, including issues in history of mathematics, the
applications of mathematics, cognitive science, etc.

APMP aims to become a common forum that will stimulate research in philosophy of
mathematics related to mathematical activity, past and present. It also aims to reach out
to the wider community of philosophers of science and stimulate renewed attention to
the very significant, and philosophically challenging, interactions between mathematics
and science. Therefore it is just natural that an affiliated meeting is being submitted to
this Congress on behalf of APMP. We asked the members of APMP to submit a proposal
for taking part in this meeting and we made an appropriate selection of submission so as
to shape a one‐day program. The aim of the meeting is to manifest the presence and
activity of APMP within the larger community of philosophers of science sand logicians.
In order to reach this aim we have opted for the format of two main lectures, followed
by shorter communications.

Main lectures:

Philippe Nabonnand, Mathematical practice from an historical point of view
James Tappenden, Richness and cognitive value in the aesthetics of mathematics

Shorter Communications:

Andrew Aberdein, Explanation and Argument in Mathematical Practice
Karine Chemla, Describing mathematical practices and the interpretation of texts.
Some views based on Chinese mathematical sources
Albrecht Heeffer, The Recreational and the Mathematical Practice of Récréations
Mathématiques (1624)
Bart van Kerkhove, Argumentative networks: Integrating philosophical approaches to
argumentation, AImodels,
and mathematical practice
Ladislav Kvasz, Rooting (Maddian) realism in mathematical practice
Javier Legris, Symbolic knowledge in Peirce’s existential graphs
Jemma Lorenat, Kronecker's Constructs
Benedikt Löwe, The interface between philosophy of mathematics and the empirical study
of the mathematical profession
Danielle Macbeth, Proof and Understanding in Mathematical Practice
Madeline M. Muntersbjorn, Real Relations between Possible Objects
Jairo da Silva, The Applicability of Mathematics

Indicative Programme Schema:

9h00‐10h00: Main Lecture, 1 [45’ talk + 15’ discussion]
10h00‐11h00: Main Lecture, 2 [45’ talk + 15’ discussion]
Break
11h15 – 11h45 Communication, 1 [20’ talk + 10’ discussion]
11h45 – 12h15 Communication, 2 [20’ talk + 10’ discussion]
12h15 – 12h45 Communication, 3 [20’ talk + 10’ discussion]
Break
14h30 – 15h00 Communication, 4 [20’ talk + 10’ discussion]
15h00 – 15h30 Communication, 5 [20’ talk + 10’ discussion]
15h30 – 16h00 Communication, 6 [20’ talk + 10’ discussion]
16h00 – 16h30 Communication, 7 [20’ talk + 10’ discussion]
Break
17h00 – 17h30 Communication, 8 [20’ talk + 10’ discussion]
17h30 – 18h00 Communication, 9 [20’ talk + 10’ discussion]
18h00 – 18h30 Communication, 10 [20’ talk + 10’ discussion]
18h30 – 19h00 Communication, 11 [20’ talk + 10’ discussion]

Abstracts

PHILIPPE NABONNAND, Mathematical practice from an historical point of view
Considerations about mathematical practice are very often linked with
problematic concerning production of mathematics. Another problem with the use
of the phrase ‘mathematical practice’ is that frequently we don’t know who is
practicing. I will try in the talk to argue why we have to take in account processes
of circulation, teaching and publishing contexts, … to grasp practical features of
mathematical activities of individual, collective or even abstract actors in social
and historical contexts.

JAMES TAPPENDEN, Richness and cognitive value in the aesthetics of mathematics
There is a standing puzzle in the epistemology of scientific theory choice. Often
apparently aesthetic criteria (beauty, elegance, ...) contribute to our choice of
theories. But why should the attractiveness of a theory be any guide to its truth?
The puzzle is reinforced by a presupposition dating back at least to Kant, holding
that aesthetic judgments are in some way independent of pragmatic advantages
("purposiveness without purpose"). I want to examine some examples of aesthetic
judgments in mathematics to bring out that at least some assessments of a piece of
mathematics as beautiful is bound up in crucial ways with problem solving
potential ("purposiveness with as yet undetermined purposes", so to speak). The
key categories of assessment are those like "richness", "fruitfulness", "intricacy".
I'll consider aesthetic judgments in two other areas ‐ music and chess ‐ to show
that mathematics is not from unique in its "non‐Kantian" aesthetic.

ANDREW ABERDEIN, Explanation and Argument in Mathematical Practice
The paper defends Kitcher’s account of mathematical explanation from criticism by
Hafner and Mancosu, by substituting higher‐level ‘argumentation schemes’ for
Kitcher’s purely syntactic ‘argument patterns’. Prospects for integrating this
approach with an independent characterization of explanation in terms of
argumentation are also examined.

KARINE CHEMLA, Describing mathematical practices and the interpretation of texts.
Some views based on Chinese mathematical sources
The paper argues that the description of mathematical practices is an essential
task for the interpretation of our sources. It is thus to be developed as a tool for
any critical approach of the documents on which we rely for a historical inquiry.
The examples will be taken in Chinese sources of the middle ages. I shall show how
the diagrams were practised in a specific way, which demands to be attended to, if
we are to understand what is at stake in the texts.

ALBRECHT HEEFFER, The Recreational and the Mathematical Practice of Récréations
Mathématiques (1624)
The book Récréations Mathématiques (1624), wrongly attributed to the Jesuit Jean
Leurechon, established a practice which has since been called ‘recreational
mathematics’. But in what sense can mathematics considered recreational and in
what way is recreational mathematics different from scholarly mathematics? We
approach these questions from the historical context of medieval and Renaissance
mathematical practices. The adjective ‘recreational’ is related to the concept of
subtlety as used in Cardano?s popular work De Subtilitate (1550). We characterize
the recreational tradition as a sub‐scientific one primarily based on oral
dissemination and open to cross‐cultural influences. By means of some typical
problems we compare recreational practices with the scholarly approach. We also
show how recreational mathematics relates to the experimental practices of the
magical tradition.

BART VAN KERKHOVE, Argumentative networks: Integrating philosophical approaches to
argumentation, AImodels, and mathematical practice
Recent studies of argument comprise three more or less separate fields: formal
argumentation theory, including dialogue games, neo‐Gricean pragmatics, and
adaptive logics; informal argumentation theory, including pragma‐dialectics and
various traditions of case‐based analysis of putative fallacies; and argumentative
networks, agent‐based models of competition between arguments, developed
primarily for use in AI. This paper offers a comparative study of these potentially
mutually inspiring approaches, with an application to mathematical practice. It is
informed by a case study of the classification theorem of finite, simple groups,
selected because it has a proof no single mathematician can comprehend in its
entirity‐‐‐it has to be `shared' by a community of mathematicians. This invites the
use of argumentative frameworks.

LADISLAV KVASZ, Rooting (Maddian) realism in mathematical practice
In her Realism in Mathematics Penelope Maddy proposed a realistic position for set
theory, which she later abandoned due to severe criticism. In the paper I will
analyze the different kinds of mathematical practice that Maddy referred to in the
articulation of her position (counting small collections of objects, applying
mathematics in physics, proving theorems about large cardinals). It turns out, that
if we make use of a richer notion of mathematical practice, the position of
mathematical realism can be effectively defended against criticism.

JAVIER LEGRIS, Symbolic knowledge in Peirce’s existential graphs
Leibniz introduced the idea of symbolic knowledge (cogitatio caeca or cogitatio
symbolica) in his work in order to draw a fundamental distinction between forms
of cognitive representations. According to this idea symbolic systems are basically
notations or writings that have as a main feature the visual displaying of structural
relations of the objects in question. The aim of this paper is to show that this
notion was implicitly in the philosophical assumptions of Peirce’s logical system of
existential graphs, where diagrams are characterized by being similar to their
objects and by being manipulated in order to obtain information concerning their
denotation. This characterization implies the visualization of signs and also actions
on them. Moreover, logical diagrams should show structural similarity with logical
forms.

JEMMA LORENAT, Kronecker's Constructs
Kronecker is historically infamous for his prohibitions against non‐constructive
mathematics. This paper will explore Kronecker's criteria for construction of
mathematical proofs and definitions. While classical examples of Euclidean
geometric construction served as metaphorical inspiration, Kronecker's essential
building blocks were the natural numbers. Within this framework, what did it
mean for something to be purely arithmetically constructed? Furthermore, how
were variables, relations, and functions built up from a natural number concept? In
his arguments against non‐constructive mathematics, Kronecker cited Gauss,
Dirichlet, and Kummer as predecessors. This paper will examine the true extent of
these claims to a constructive tradition as well as the refinement of Kronecker's
position in reaction against non‐constructive methods and how these new
methods succeeded in wide acceptance.

BENEDIKT LÖWE, The interface between philosophy of mathematics and the empirical study
of the mathematical profession
The recent naturalistic turn of philosophy of mathematics has resulted in renewed
interest of philosophers in the study of mathematical activity and practice. This is a
field studied in depth by researchers in mathematics education and to some extent
by researchers in STS (Science and Technology Studies). But which parts of this
field are philosophically relevant? The talk will describe a few case studies of how
philosophers can fruitfully interact with the research done in the other fields.

DANIELLE MACBETH, Proof and Understanding in Mathematical Practice
The mathematical practice of proving theorems seems clearly to result in
improved mathematical understanding; the aim and point of proving, and
reproving, theorems in mathematics is better mathematical understanding. And
yet, it has become increasingly clear that proof as it is usually understood (a proof
is a deduction of a theorem on the basis of axioms) is irrelevant to understanding.
There are only two options: either mathematical understanding resides
somewhere else than in proof, or a mathematical proof is something different from
what the standard conception says it is. Both options have been pursued: Manders,
for instance, arguing that mathematical understanding resides not in proofs per se
but in the conceptual settings that mathematicians develop and deploy in their
proofs, and Rav, among others, arguing that the mathematician’s proofs are
essentially different from proofs as they are understood in mathematical logic. In
such a circumstance, one would like to “go between the horns”. I will suggest how
that might be done.

MADELINE M. MUNTERSBJORN, Real Relations between Possible Objects
This essay challenges interpretations of Poincaré that contrast his
“conventionalist” philosophy of science against his “Kantian” philosophy of
mathematics by noting the influence of evolution on his thinking about mind and
world in general. Like natural kinds, what we know about the world is neither
inert nor eternal, but comes into being gradually over time and is constrained by
what has come before. To understand how knowledge grows, we must understand
how the universal mind of the logician and the inherited mind of the psychologist
together enable us to see real relations between possible objects. By attending to
the different representational systems employed during phases of discovery,
including arbitrary symbols, genetic algorithms, and action schemas, we offer a
more integrated interpretation of Poincaré’ as well as a more contemporary
account of those practices that contribute to the growth of knowledge over time.

JAIRO DA SILVA, The Applicability of Mathematics
Some scientists and philosophers (most notably Wigner and Steiner, of course)
believe that the "wonderful" fact that mathematics is applicable in science is either
inexplicable or explicable only by placing man in a special relation to nature. In my
talk I plan to argue against this mystical trend by offering a much more mundane
account of the scientific applicability of mathematics. I will, in particular, show
some of Steiner's favorite examples of the heuristic value of (supposedly) formal
analogies (Maxwell's discovery of displacement currents and Dirac's discovery of
anti‐matter, for example) under a different, more revealing, light.