**Abstracts**

**----------------------------------------------**

James C. Alexander (Case Western Reserve)

*Cognitive structures and semiotics of mathematics*

We explore how mathematics--higher mathematics--has incorporated
cognitive mechanisms into its formal structure, a fact that is relevant
to the continuing fecundity of the discipline. From this, we segue
to (preliminary) questions of how the structres are implemented,
and in particular, what of the semiotics of mathematics? That is,
what does the formalism of mathematical notation tell us about mathematical
cognition?

**----------------------------------------------**

**Myrdene Anderson, **Purdue (Anthropology
and Semiotics)

*Qualification of Quantification across the Curriculum*

At least in the social and behavioral sciences-so delineated respecting
the preference from psychology-practitioners and students alike
have subscribed to another distinction, that between qualification
and quantification. In fact, one might easily get the impression
that the latter distinction amounts to a debate, if not outright
warfare.

Students arrive to their disciplines largely ignorant of the semiosic
underpinnings of their own intellectual experiences and of the semiotic
foundations for their elected fields of study. In addition, typically
without a broad background in the history and philosophy of the
disciplines, students fail to perceive how all of them interlock
in a myriad of ways. Being so undisciplined about their disciplines,
it is not surprising how students can be so easily recruited. The
majority subscribe to a contrast between qualification and quantification,
and usually in so doing they privilege the latter, assuming it to
be the more "scientific" approach for meaning-making.
At the same time, though, the condition of relative innumeracy of
young and old, within and beyond any but the most technical fields,
has been conceded without a struggle. As a consequence an unhealthy
hierarchy is fractured by feelings of inadequacy and/or resentment.

Perhaps with greater engagement in public education in the future,
affording younger students trickling up through the ranks a more
inquiring interest in the qualia and quanta across the cultural
curricula, higher education may enjoy a new lease on intellectual
work and play. Meanwhile, the position taken in this paper asserts
that both qualification and quantification are cultural practices,
and further that "etic" quantification is predicated on
prior "emic" qualification (drawing on Kenneth L. Pike).
The "cultural" may at the social level include the linguistic
and at the individual level include both bodily and cognitive processes.

At the same time, higher education may be able to benefit from
considering approaches aimed at younger audiences. These include
the "Singapore method"; the "Private Eye" approach
to looking and thinking by analogy; the foregrounding of abduction
larding the more recognized induction and deduction; and building
on awareness through movement.

**----------------------------------------------**

**Joachim De Beule, **Free University
of Brussels

*Agent-Based and Mathematical modeling in Semiotic Dynamics*

Semiotic dynamics is the field that studies the dynamics associated
with the formation, the usage and the evolution of semiotic systems.
It grew out of research in artificial intelligence and evolutionary
linguistics, and originally was mainly concerned with human (and
human- like) languages [6, 4]. However, there is a growing body
of evidence that coding and semiosis are fundamental ingredients
of all life. This recently lead to a new scientific field called
biosemiotics, defined as the study of signs, of communication, and
of information in livingorganisms, and claimed to provide a new
understanding of life [1]. If semiosis is fundamental to life, then
a proper understanding of semiotic dynamics will be crucial in order
to fully understand all aspects of macro-evolution, in particular
the appearance of new codes and new levels of selection during major
transitions [5, 2].

In this talk, I will review some of the basic notions in (coding)
biosemiotics and semiotic dynamics and show how, in recent years,
researchers from a variety of fields were able to, make significant
progress through a combination of multi-agent simulations and mathematical
(formal) modeling. I will focus on the notion of language games
[7]. These are used in semiotic dynamics as vehicles for the study
of evolution through conventionalization, much in the same way as
games are used in evolutionary game theory to study evolution through
natural selection. In particular, I will focus on the naming game,
one of the simplest and currently best understood language games
[3]. It will be shown how the naming game dynamics can be analyzed
mathematically and can give rise to phase transitions in the coding
behavior of individual code users corresponding to an increased
state of global coordination. Some possible applications and extensions
of the naming game will be discussed and a number of open problems
will be identified.

References

[1] Marcello Barbieri. Biosemiotics: a new understanding of life.
Naturwissenschaften 95(7), pages 577–599, July 2008.

[2] Marcello Barbieri. The mechanisms of evolution. natural selection
and natural conventions. In Marcello Barbieri, editor, The Codes
of Life, chapter Dordrecht, pages 15–35. Springer, 2008.

[3] Andrea Baronchelli, Vittorio Loreto, and Luc Steels. In-depth
analysis of the naming game dynamics: The homogeneous mixing case.
International Journal of Modern Physics C, 19(5):785–812, May
2008.

[4] Ciro Cattuto, Vittorio Loreto, and Luciano Pietronero. Semiotic
dynamics and collaborative tagging. Proceedings of the National
Acadamy of Sciences, 104(5):1461–1464, January 2007.

[5] J. Maynard-Smith and E. Szathm´ary. The major transitions
in evolution. Morgan-Freeman, 1995.

[6] Luc Steels and Frederic Kaplan. Collective learning and semiotic
dynamics. In Floreano, Nicoud, and Mondada, editors, Advances in
artificial life, Proceedings of ECAL’99, volume 1674 of Lecture
Notes in Computer Science, 1999.

[7] Luc Steels and Paul Vogt. Grounding adaptive language games
in robotic agents. In Phil Husbands and Inman Harvey, editors, Proceedings
of the Fourth European Conference on Artificial Life (ECAL’97),
Complex Adaptive Systems, Cambridge, MA, 1997. The MIT Press.

**----------------------------------------------**

**John Deely, **St. Thomas University

*The Semiosis of Mathematical Thinking*

This paper discusses mathematical thinking only insofar as that
thinking provides an unmistakable prime example of anthroposemiosis
in its species-specific difference from all the varieties of zoösemiosis.
Thus, recurring to Euclid’s triangle as a central example,
my aim is to outline how relation as a mode of being exhibits a
singularity that proves to be the basis for the prior possibility
of semiosis in general, a singularity that mathematical objectivity
makes particularly recognizable even though the feature in question
extends to the full range of semiosis as an action transcending
the contrast between mind-dependent and mind-independent being.

**----------------------------------------------**

**Keith Devlin**, Stanford (Mathematics)

*Using a video game as a medium to represent mathematics for children
learning basic mathematics*

To date, almost all video games designed to help students learn
mathematics do little more than present traditional mathematics,
represented by standard symbolic expressions, into a video game
wrapper. In essence, they regard video games as a new medium on
which to pour symbols. But video games provide an entirely new way
to represent mathematics. I have spent the past five years investigating
how to make use of the natural affordances in video games to do
just that.

**----------------------------------------------**

**Vitaly Kiryushchenko**, St. Petersburg
State School of Economics

*MAPS AND SIGNS: THE VISUAL AND THE VIRTUAL IN PEIRCE'S SEMIOTICS*

In May 1879 (the year Peirce's careers as a scientist and academic
philosopher first overlapped, as he started teaching at the Johns
Hopkins while continuing his research for the US Coast Survey),
Peirce published a short paper in the American Journal of Mathematics
describing his new map projection, which he called "quincuncial".
The quincuncial map was a variation of conformal stereographic projection
and, besides, one of the first diagrammatic pictures created with
the application of complex analysis.

The present paper, by using an example of Peirce's quincuncial map,
is aimed at showing how some of Peirce's late pragmatist and semiotic
ideas were developed from his early practice as a scientist and
a mathematician, thus providing an intriguing example of the intersection
of scientific practice and philosophical speculation.

**----------------------------------------------**

**Kalevi Kull**, University of Tartu, Estonia
(Biology and Semiotics)

*What mathematical structure is semiosis (if any at all)?*

1. Semiotics and mathematics are highly incompatible - likewise
poetry and logic, or life and automaton.

2. There exists a long and rich tradition of modelling of some
semiosic objects, as well as a search for proper mathematical tools
for thier modelling, e.g., of organisms and languages. Yet, mathematical
biology and mathematical linguistics (also mathematical sociology,
mathematical psychology, etc.) have challenged the problem of limitedness
of mathematization of their theoretical core. In its general form,
this is the problem of mathematical description (modelling) of semiosis.
It will be instructive to review some examples about the searches
for mathematical description (formalization) of semiosis (e.g.,
of Robert Rosen, of computational semiotics, of algebraic semiotics,
etc.).

3. A feature that is in the focus of modelling of semiosis has
to be the feature that natural languages possess and the formal
languages (and formal logic) do not. If this feature resists formalization,
then how is it possible that mathematics can describe the physical
world, whereas it (as if) cannot describe the non-mathematical or
natural language from which it is a derivative?

4. Modelling of semiosis appears to be particularly a problem for
biosemiotics. Because if to accept Sebeok's thesis that semiosis
is the criterion of life, then a model of sign is simultaneously
a model of life. Thus the problem of distinction between natural
and formal languages goes beyond languages. If languages are defined
as systems that use symbols, then there exist many semiosic systems
that are not languages, i.e. which include merely non-symbolic semiosis
(usually called non-human life forms). This is, in other words,
an old question whether it is possible to distinguish between living
and non-living system, i.e. between informal (or natural) and formal
sign systems on the basis of their formal (mathematical) descriptions.

5. Formal sign systems are derivatives from natural sign systems
- similarly to artefacts and dead languages. What makes them formal
is the lack of the very feature the modelling of semiosis is addressing
- the semiosity, the life itself. The existence of codes is a necessary
but not a sufficient condition for semiosis, because constructions
of non-living machines (i.e., certain artefacts) also include codes.

6. The sciences that are dealing with everything that can be described
in an unambiguous (formal) way can be called phi-sciences (physical
sciences), whereas the sciences that can deal with equivocal (polysemous,
like natural language, poetry and life itself) descriptions can
be called sigma-sciences (semiotic sciences).

**----------------------------------------------**

**Solomon Marcus**, Mathematical Section
of the Romanian Academy (Mathematics)

*Mathematics, Between Semiosis and Cognition*

1. The nature of mathematical cognition is controversial.

2. The invention-discovery interplay in mathematics raises delicate
questions.

3. Is Mathematics, like Music, predominantly syntactic?

4. Mathematical rigor is frequently obtained at the expense of meaning.

5. Mathematical concepts emerge from diaphoric self-referential
metaphors.

6. Mathematics of the macroscopic universe is associated with human
language and semiosis, relatively sharp distinction between subject
and object, Euclidean paradigm and Galileo-Newtonian paradigm.

7. The mathematics of the infinitely small and of the infinitely
large adopts the strategy of Plato's allegory of the cave.

8. Cognitive mathematical models and metaphors are by their nature
conflictual.

**----------------------------------------------**

**Yair Neuman, **Ben-Gurion University of
the Negev

*Semiotics, Mathematics & Information Technology: The Future
is already here ...*

Semiotics is considered as an anachronistic academic field the same
as Philology and Egyptology. In this presentation, I would like
to refute this dogma and to show how the synergy of Semiotics, as
a meta-perspective for cognition, Category Theory as a meta-perspective
for mathematics, and Information Technology as a meta-tool for computation,
may introduce breakthroughs in the study of cognition.

Based on our latest research and algorithms, I would like to review
some of these breakthroughs: How we may automatically screen for
depression through the use of metaphors, an algorithm that differentiates
between denotation and connotation (wet hair vs. wet dream) and
identifies the meaning of an abstract connotation (wet dream = erotic
dream), how "Hypostatic Abstraction" proposed by Peirce
explains the way "language" enables the abstraction of
"thought", and how semiotic ideas may be used for semi-automatically
excavating hidden and unconscious themes in group-dynamics.

**----------------------------------------------**

**Frank Nuessel**, University of Louisville
(Semiotics and Linguistics)

*The Representation of Mathematics in the Media*

Mathematics and mathematical concepts appear in the print(books,
newspapers, magazines, other publications, text-based objects) andnon-print
(television and cinema) media with some frequency. This paperexamines
selectively these popular cultural manifestations of mathematics
withspecial attention to those exemplars that incorporate reasonably
accurateversions of mathematics. A limited number of copies of the
text of the paperwill be available for the audience.

**----------------------------------------------**

**Rafael Nunez, **University of California,
San Diego

*What is the nature of mathematics? A view from the Cognitive
Science of the number line*

Mapping numbers to space is fundamental to mathematics. The number
line is arguable one the simplest but richest examples of the power
of such mappings. But, what are its cognitive origins? Are the intuitions
underlying the number line "hard-wired"? Is the number
line a cultural construct? Contemporary research in the psychology
and neuroscience of number cognition has largely assumed that the
representation of number is inherently spatial and that the number-to-space
mapping is a universal intuition rooted directly in brain evolution.
I'll review material from the history of mathematics as well as
empirical results from two of our recent studies to defend a radically
different picture: the representation of number is not inherently
spatial and the intuition of mapping numbers to space is not universal.
In one study we show that there are non-spatial representations
of numbers that co-exist with spatial ones, as indexed by instrumental
manual actions, such as squeezing and bell-hitting, and non-instrumental
actions, such as vocalizing. Moreover, the results suggest that
the number-to-line mapping—a *spatial* mapping— is not
a product of the human biological endowment but that it has been
culturally privileged and enhanced. The other study, which we carried
out in the remote mountains of Papua New Guinea, shows experimentally
that individuals from a culture that has a precise counting system
(and lexicon) for numbers greater than twenty lack the intuition
of a number-to-line mapping, suggesting that this intuition is not
universally spontaneous, and therefore, unlikely to be rooted directly
in brain evolution. The number-to-line mapping appears to be learned
through— and continually reinforced by— specific cultural
practices, such as measurement tools, writing systems, and elementary
mathematics education. It is over the course of exposure to these
cultural practices that well-known brain areas such as the parietal
lobes are recruited to support number representation and processing,
which in turn, allow the learning of more elaborated mathematical
concepts.

**----------------------------------------------**

**Wolf-Michael Roth**, University of Victoria
(Semiotics and Mathematics)

*Tracking the Origin of Signs in Mathematical Activity: A Material
Phenomenological Approach*

The semiotic literature tends to take signs as given. Even in constructivist
and embodiment accounts of cognition, the sign, such as a gesture
that exhibits some linear relation or trend, is merely the enactment
of a pre-existing schema the enactment of which results in the production
of a sign. Yet empirical evidence shows that the human sign form,
as thing that stands for another thing, does not constitute the
beginning – children do not make a distinction between the
thing and its name. To understand signs, we therefore need to take
a genetic perspective. In this paper, I present a material phenomenological
account of how signs and thoughts emerge in ongoing activity. I
provide a very detailed description of a lecture excerpt, essential
features of which cannot be explained by presupposing signs, thoughts,
or mental schema. The approach I offer provides explanations for
some of the difficult problems in education, psychology, and cognitive
science.

**----------------------------------------------**

**Fernando Soto**

*Lewis Carroll's Name-Number Nexus: Snarky Wonderland Games.*

In this paper I first set out to demonstrate how the etymologically-trained
mathematician Charles Lutwidge Dodgson (better known as Lewis Carroll)
uses ambiguity as he confuses and abuses words and their meanings
in his "Nonsense" works. I then trace the etymological
links between the words "name," "number," and
"numismatic," and how Carroll creatively used and confused
them. I then go on to provide several example of name/number/coining
word-play in Alice's Adventures in Wonderland, The Hunting of the
Snark, and Through the Looking-Glass. I finish off the talk by presenting
a partial etymological answer to Carroll's famous "Why is a
raven like a writing-desk" riddle.

**----------------------------------------------**

**Kumiko Tanaka-Ishii**, University
of Tokyo, Japan (Computer Science)

*Dyadic vs. Triadic Sign Models Through Computer Programs*

The theme of this talk is the relation between the dyadic and triadic
sign models in the context of computer programs as the semiotic
target. The content of the talk appeared in my recent book "Semiotics
of Programming," published by Cambridge University Press in
2010. Since computer programs are mechanically interpreted on machines
and are therefore rigorous, semiotic analysis of programs enables
formal reconsideration of the sign models proposed so far.

Among various dyadic and triadic models, the discussion here centers
on the correspondence between the triadic sign model proposed by
Peirce and the dyadic sign model proposed by Saussure. Traditionally,
it has been thought that Peirce's interpretant corresponds to Saussure's
signified, and that Saussure's model lacks Peirce's object. Analysis
of the two most widely used computer programming paradigms, however,
suggests that Peirce's object formally corresponds to Saussure's
signified, and that Saussure's sign model is obtained when Peirce's
interpretant is located outside his model in the programming language
system. Further, I suggest how this distinction may dissolve when
signs are introduced through self-reference.

**----------------------------------------------**

**Mark** **Turner, **Case Western (Cognitive
Science)

*Mental Packing and Unpacking in Mathematics*

Recently, it has been hypothesized that human working memory greatly
increased during the Upper Paleolithic, and that this evolutionary
change caused, or at least sparked, the cognitively modern human
mind, with its outstanding creative capacities, including mathematical
insight and scientific discovery. According to this “working
memory” explanation of our unusual abilities, capacious working
memory made it possible to activate large ranges of ideas and connect
them creatively; it produced conceptual networks much more complicated
than anything that had previously been possible (Wynn 2002, Wynn
& Coolidge 2003 & 2004, Wynn, Coolidge, & Bright 2009,
Balter 2010). Be that as it may, the management of complex conceptual
networks requires quite different powers of conceptual packing and
unpacking. Packing and unpacking make working memory useful. Packing
and unpacking are provided by the basic mental operation of conceptual
integration, otherwise known as blending. Blending provides packed
mental structures congenial to human cognition. These packed mental
structures can be carried without the assistance of working memory,
and later unpacked creatively to create large conceptual networks
involving new information. In this way, the packed blend provides
something that can travel with us mentally for future service. The
packed blend is additionally indispensable in providing a congenial
basis, suited to human cognition, from which to grasp and manipulate
conceptual networks that would otherwise lie far beyond our ability
to maintain, manage, and manipulate. Mathematics as we know it is
in large part made possible by this feature of conceptual integration.
Mathematics specializes in providing instruments of packing and
unpacking.