Mathematical Structures of Natural Intelligence

168 Pages·2017·1.481 MB·english

by  Yair Neuman

Yair Neuman
Mathematical Structures
of Natural Intelligence
Yair Neuman
The Department of Brain and Cognitive Sciences
and the Zlotowski Center for Neuroscience
Ben-Gurion University of the Negev
Beer-Sheva, Israel
ISSN 2522-5405 ISSN 2522-5413 (electronic)
Mathematics in Mind
ISBN 978-3-319-68245-7 ISBN 978-3-319-68246-4 (eBook)
https://doi.org/10.1007/978-3-319-68246-4
Library of Congress Control Number: 2017957645
© Springer International Publishing AG 2017
Preface
For many years, I have been interested in identifying general struc-
tures underlying the various expressions of the human mind, from
psychology to art and language. This interest has naturally led me to
an intensive reading of texts such as those of the gestalt psychologists
and of Jean Piaget and Gregory Bateson, whose interdisciplinary
approach has had a great influence on my work. The idea of writing a
book on “mathematical structures” of “natural intelligence” has
slowly grown in my mind, always accompanied by two main
concerns.
The first concern is that, in the postmodern and post-structuralist
phase of the Western “intelligentsia,” such a venture might be consid-
ered both anachronistic and pretentious. I have tried to address these
concerns first by presenting a neo-structuralist approach that addresses
some of the difficulties that were associated with the old structuralist
venture and second by putting aside unjustified pretensions toward
developing a grand theory. In contrast with some such grand theories of
the past, this book is framed from a much more reflectively humble
perspective.
The second major concern that bothered me is that, in the context
of studying the human mind, “mathematical structures” may be a nice
metaphor in some cases but it can also be an empty abstraction.
Learning about past attempts to “mathematicize” the human mind,
such as those of Kurt Lewin and his field theory, has left me with a
strong and sour taste of disappointment. Mathematics is such an
abstract field that, beyond the powerful technical tools it provides for
outsiders, its ability to provide informative structures for modeling
the human mind is extremely limited. We may enjoy ideas such as
“topological psychology,” but these intellectual games seem to lead
nowhere.
In this book, I use the term “mathematical structures” to describe in
an abstract language several configurations that may be highly infor-
mative in illuminating some aspects of our mind in fields ranging
from neuroscience to poetry. The reader may therefore not find herein
any attempt to introduce the “mathematics of natural intelligence.”
However, I still aim to identify general abstract structures that may be
used for modeling “natural intelligence” (a term that is briefly intro-
duced below and then clarified in Chap. 1).
Having said that, and being highly cognizant of the limits of a neo-
structuralist venture, the final decision to write the book crystallized
in my mind when I visited the beautiful city of Naples for a confer-
ence. One night, and while sitting near the window of my hotel room
and observing the sea with its repeating patterns of waves, some
threads wove together in my mind. I suddenly realized how deeply
interconnected are several ideas that have interested me for many
years, from Russell’s definition of a number to the mathematical con-
cept of the groupoid. From there, this book naturally emerged in the
form you are currently reading.
Given this context, the book’s organization is as follows. The first
part of the book introduces the justification for studying general struc-
tures that we may use to model “natural intelligence.” The term “intel-
ligence” is used in the broad sense of computing patterns, and the
adjective “natural” is used to draw a boundary between intelligence as
it is evident in natural living systems and intelligence as it is studied
in artificial systems and models. In fact, computation is the leading
metaphor for studying the mind (Crane, 2015), and the inevitable
question is: What is the difference between mathematical and compu-
tational modeling of the mind? This is quite a serious question that
deserves an in-depth discussion which is beyond the scope of this
book. However, while computational models of the mind attempt to
understand human cognition in metaphorical/analogical terms of an
abstract Turing machine, and/or to produce computational models of
specific cognitive processes, in this book, I model human cognition by
drawing on the very abstract theorization of category theory.
Differences and similarities between computational models and the
mathematical models presented in this book can be easily found, but
the important point is that my theoretical point of departure for under-
standing the human mind is totally different.
I present Piaget’s heroic venture to lay the grounds for a theory of
structure but critically point to his failure. To address Piaget’s failure,
I introduce and use the language of category theory, a field of mathe-
matics that has great relevance as a powerful tool for building models.
The book is self-contained, in that it doesn’t assume any prior knowl-
edge of mathematics in general or of category theory in particular.
However, I am aware of the fact that the book is extremely challeng-
ing in its level of theorization and abstraction and that it will require
effort from the reader to struggle with some of its abstract formula-
tions. I believe, though, that the intellectual benefits of these efforts
are justified. Using category theory, I argue that a certain mathemati-
cal structure – the groupoid – may be used to address Piaget’s main
problem and to serve as a building block of structure. To illustrate the
explanatory power of this conceptualization, I use the field of neuro-
science and explore an unresolved question, which is why we observe
local and recurrent cortical circuits in the mammalian brain.
The second part of the book aims to take us a step further by delv-
ing deeper into the power of category theory’s conceptualization in
modeling structure. I explain why natural intelligence is deeply “rela-
tional” and how structures such as natural transformation may model
these relational processes. Moreover, I explain that natural intelli-
gence is not only relational but also value laden and explore the gestalt
aspect of structure by using a variety of category theory tools. In this
part, I also introduce a novel principle that I describe as the natural
transformation modeling principle and explain how it can serve as an
alternative concept for studying uncertainty “in the wild.”
The third and concluding part of the book is the least mathematical
and involves the application of the general principles and ideas pre-
sented in the first two parts, in order to reach a better understanding of
the three challenging aspects of the human mind. In this part of the
book, I try to explain the human representation of the number system
and specifically why our counting ability is different from the ability
evident among non-human organisms and why it is so difficult to
grasp the idea of zero. I also model the process of analogical reason-
ing and metaphor by pointing to its underlying relational structure and
its deep grounding in episodic memory and in cultural semiotic
threads necessary for understanding the complexity of metaphors.
Taking this idea a step forward, in this third section of the book and
following the theory of Ignacio Matte-Blanco, I adopt the idea of the
“unconscious” as expressing creative processes of symmetrization
and illustrate how deeply connected these processes are to the general
themes presented in the book and how they may be used to broaden
our understanding of metaphorical processes and the creativity of the
human mind.
As the reader may have already realized, the book is intellectually
challenging, and for this reason, only the efforts required to cross the
bridge of mathematical abstraction seem to be fully justified. To ease
the cognitive load the reader may experience, I have added a bullet
summary at the end of each chapter. In addition, I have not overloaded
the reader with references or footnotes, which are sometimes used to
express the author’s “deep mastery” in his academic field.
An etymology1 of “mastery” in the English language explains that
mastery appeared as a transitive verb (i.e., a verb with a defined object)
in 1225 and was used in the sense of “overcoming,” in the military
sense of defeating an enemy. The learned scholar, the “úþwita” (Old
English), who is beyond (i.e., úþ) the ordinary wisdom of human
beings is therefore the one who “defeats” and “conquers” the object
of knowledge. But can we really conquer knowledge?
The old Jewish rabbis who read the book of Ecclesiastes encoun-
tered the following text: “All the rivers run into the sea, yet the sea is
not full; unto the place whither the rivers go, thither they go again.”
They metaphorically interpreted “rivers” as “wisdom” and “sea” as
corresponding to one’s heart. According to this interpretation, one
may gain wisdom, but wisdom cannot be conquered, defeated, or
“mastered” as if it were a beast to be tamed. Throwing the metaphor
of mastery aside, we may step into the river.
Beer-Sheva, Israel Yair Neuman
1 All references to the etymology of words are drawn from The Historical Thesaurus of English
unless stated otherwise.
Contents
Part I
1 I ntroduction: The Highest Faculty of the Mind . . . . . . . 3
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 W hat Is Structure? Piaget’s Tour de Force . . . . . . . . . . . 13
Piaget on Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Reversibility and Irreversibility . . . . . . . . . . . . . . . . . . . . . . 25
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3 C ategory Theory: Toward a Relational Epistemology . . 31
Universal Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
The Co-product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4 H ow to Trick the Demon of Entropy . . . . . . . . . . . . . . . . 47
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5 N eural Networks and Groupoids . . . . . . . . . . . . . . . . . . . 53
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
Part II
6 N atural Intelligence in the Wild . . . . . . . . . . . . . . . . . . . . 65
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
7 N atural Intelligence Is About Meaning . . . . . . . . . . . . . . 71
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
8 From Identity to Equivalence . . . . . . . . . . . . . . . . . . . . . . 77
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
9 O n Negation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
10 M odeling: The Structuralist Way . . . . . . . . . . . . . . . . . . . 93
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
11 O n Structures and Wholes . . . . . . . . . . . . . . . . . . . . . . . . 103
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
Part III
12 Let’s Talk About Nothing: Numbers and Their Origin . 121
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
13 King Richard Is a Lion: On Metaphors and Analogies . 131
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
14 T he Madman and the Dentist: The Unconscious
Revealed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
15 D iscussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
References ......................................... 163
About the Author. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
Figures
Fig. 2.1 Smiley’s face ............................................................... 16
Fig. 2.2 Smiley’s disorganized face ......................................... 16
Fig. 3.1 A map illustrating the permutation of three boxes ..... 32
Fig. 3.2 A structure with three isomorphic objects .................. 38
Fig. 3.3 Projections of A × B ................................................... 39
Fig. 3.4 The commuting diagram ............................................. 39
Fig. 3.5 The taxonomy of Chair ............................................... 40
Fig. 3.6 The product as a pattern .............................................. 41
Fig. 3.7 The co-product ............................................................ 41
Fig. 3.8 The Würstchen and the Hotdog .................................. 45
Fig. 4.1 The product’s projections ........................................... 48
Fig. 4.2 Product and identity .................................................... 48
Fig. 4.3 The co-product of sets ................................................ 48
Fig. 4.4 The proof .................................................................... 48
Fig. 4.5 The groupoid ............................................................... 49
Fig. 4.6 A two-cell network ..................................................... 49
Fig. 6.1 An apple in transformation ......................................... 67
Fig. 6.2 The first three phases of consumption ........................ 67
Fig. 6.3 The consumption as a groupoid .................................. 67
Fig. 6.4 A functor between the transformations functors ........ 68
Fig. 7.1 The mapping between the differences and value
categories .................................................................... 74
Fig. 8.1 Functors between the apples ....................................... 79
Fig. 8.2 A functor between functors F and G .......................... 80
Fig. 8.3 Natural transformations .............................................. 81
Fig. 8.4 Two mappings from the first appearance
of the apple ................................................................. 82
Fig. 8.5 Adjointness ................................................................. 84
Fig. 8.6 Adjointness in Köhler’s hen experiment ..................... 84
Fig. 9.1 The sub-object diagram .............................................. 88
Fig. 11.1 A dragonfly ................................................................. 105
Fig. 11.2 Sub-objects of the dragonfly ....................................... 106
Fig. 11.3 The center of the dragonfly ......................................... 107
Fig. 11.4 A checkerboard ........................................................... 108
Fig. 11.5 A rotating disk ............................................................ 111
Fig. 11.6 A diagram of US elections .......................................... 112
Fig. 11.7 The eye ....................................................................... 117
Fig. 12.1 The formation of a number ......................................... 125
Fig. 13.1 The structure of analogical reasoning ......................... 134
Fig. 13.2 The formation of a metaphor ...................................... 144
Fig. 14.1 A triangle .................................................................... 152