[FOM] Book appearing: Formalization without Foundationalism: Model Theory and the Philosophy of Mathematical Practice

[John Baldwin]

I take this opportunity my about to be published book.

Bound copies of the proofs of the above book will be available at the
Cambridge Press booth at the annual Joint Mathematics Meetings in San
Diego.  The book will be available a bit later.

Here is a lightly edited version of the book proposal.

In the last decades, a distinct area of the philosophy of mathematics has
arisen, the philosophy of math-ematical practice. This movement rejects the
notion that the only connections between philosophy and
mathematics occur through the analysis of the foundations of a global
theory of all mathematics. One reason for this rejection is the realization
that ‘reduction to set theory’, while a good tool for studying reliability,
does not preserve either the meaning or the spirit of the various areas of
mathematics. In this field, the focus shifts from reliability to
clarification; ontology takes a less central role. Research focuses on
questions of methodology, visualization, fruitfulness, explanation, and so
on. Thus, much of the literature in the area consists of historical case
studies of specific areas of mathematics.The crucial epistemological issue
between these schools is the one/many divide: should there be one
foundational scheme for all mathematics or should one undertake the study
of various areas of mathematics and develop schema for understanding the
relations between areas, relations that depend on their content(but as we
will see in a way that can be formalized)?

We argue that the paradigm shift in model theory described below, which at
first glance appears to be only technical, actually represents this
paradigm shift in epistemology. Further, it provides a principled way to
distinguish the ‘tame’ from the ‘wild’ in mathematics and provide
sufficient conditions for tameness.

We pause to explain the ‘wild’/‘tame’ distinction, which is
made commonly by mathematicians, and its connection with the philosophy of
mathematics.Formal Logic emerged around the turn of the twentieth century
in the midst of the foundational crisis. It arose as a tool for dealing
with issues of reliability in mathematics. Cantor understood a distinction
between the absolute infinite and the infinite, but Frege’s introduction of
a precise syntax revealed a paradox.
The resolution required both Zermelo’s insight in proposing the axiom of
separation and the formalization of this notion as a first order axiom of
ZFC. Gödel analyzed the liar paradox more deeply and arrived
at his incompleteness theorems. In time, the behavior of arithmetic he
isolated, undecidability and the inability (because of the pairing
function) to distinguish the various Cartesian products of N as spaces of
different dimension, has been recognized by mathematicians as wild
behavior. Wildness doesn’t impinge on mainstream mathematical
investigations; the structures mathematicians use to investigate number
theory, the real, complex and p-adic fields are tame, i.e. don’t share the
Gödel phenomena. One goal of this book is
to explain this divergence.


Between 1920 and 1970 four areas of logic: proof theory, recursion theory,
model theory and set theory developed substantial technical methods
differentiating them from the others. They retained clear philosophical
roots: reliability for proof theory and set theory and analysis of the
intuitive notion of computability for
recursion theory and of the concept of truth for model theory. Through the
1950’s both recursion theory and proof theory retained strong links with
the Hilbert program. Set theory was viewed as the foundation for
all mathematics. Model theory in the 1960’s was seen as a philosophical
tool both because of the central role of the study of models of set theory
(modern set theory is often described as the model theory of ZFC) and for
its applications in such areas as linguistic philosophy, possible world
semantics, and model theoretic
analysis of truth.

Technical developments (priority method, forcing, stability theory) in the
last three areas has led to a detachment of these areas from philosophy
(less so for set theory). In particular, the sophisticated techniques
arising in the 1960’s and 1970’s in pure model theory and the development
of model theoretic algebra have been perceived as mathematical topics
largely disjoint from concerns of mathematical philosophy. We challenge
that perception.

We call the revolution in model theory that crystalized in the early 1970’s
a paradigm shift to emphasize that the new technology arose because of a
change in approach to the fundamental problems. In short, the paradigm
around 1950 concerned the study of logics; the principal results were
completeness, compactness, interpolation and joint consistency theorems for
first order and other logics. Various semantic properties of theories were
given syntactic characterizations but there was no notion of partitioning
all theories by a family of properties.
After the paradigm shift there is a systematic search for a finite set of
syntactic conditions which divide first order theories into disjoint
classes such that models of different theories in the same class have
similar mathematical properties. Thus, we take Shelah’s stability hierarchy
[She70] as a marker of the transition.

This book is a response to these changes of direction in mathematics and
philosophy. It aims to show how the themes and methods of contemporary
model theory contribute to the philosophy of mathematical practice. At the
same time, the book itself is a case study of the sort referred to before:
the extraordinary interaction of model theory with traditional mathematics.
It presents the views on methodology and philosophy of leading
mathematicians and model theorists.

The principal argument of the book is that the process and result of
formalizing parts of mathematicsprovides tools not only for investigating
the major question raised above but for other epistemological issues
such as clarity. Further it provides new tools for mathematics.
For this study, we need more precise definitions. A formalization of a
mathematical area specifies a vocabulary (roughly, language to represent
the primitive notions of the area), a logic, and the axioms (postulates in
the technical sense of Euclid) for the particular topic.

While not denying Corfield’s argument (page x [Cor03]) that an ‘area’ is
much more than an axiomatic system, we argue both abstractly and by many
examples that the formalizing an area is often an important tool for
progress in that area.

This general theme is developed in four theses.
Theses.
1. Contemporary model theory makes formalization of specific mathematical
areas a powerful tool to in-
vestigate both mathematical problems and issues in the philosophy of
mathematics (e.g. methodology,
axiomatization, purity, categoricity and completeness).
2. Contemporary model theory enables systematic comparison of local
formalizations for distinct math-
ematical areas, in order to organize and do mathematics and to analyze
mathematical practice.
3. The choice of vocabulary and logic appropriate to the particular topic
are central to the success of a
formalization. The logic which has been most important for the study of
mathematical practice is first
order.
2
4. The study of geometry is not only the source of the idea of
axiomatization and many of the fundamental
concepts of model theory, but geometry itself plays a fundamental role in
analyzing the models of tame
theories.

The first thesis has two aspects: model theory as a mathematical topic and
tool (pure and applied model theory) and model theory as a philosophical
tool. The second thesis differs from the first in emphasizing that model
theory provides an organizing schema connecting areas of mathematics for
the needs of both
mathematics and philosophy.

Many important studies in the philosophy of mathematical practice have
deeply investigated mathematics through the early twentieth century. This
book addresses both model theory as a current research area
in mathematics and its applications in such fields as number theory,
diophantine geometry, real algebraic
geometry, and analysis. As such it is one of the first contributions to the
study of current mathematical practice. These applications are a
consequence of the organization of first order theories by the syntactic
stability hierarchy (supplemented by o-minimality). This schema provides a
systematic approach to converting informal analogies between disparate
areas of mathematics into metatheorems.
This new schema is a rich source of philosophical opportunity. Where
previously one had a rough classification into first order and second
order, and a collection of results involving completeness, categoricity,
and so on, one now has an overarching classification that serves as a
powerful tool for epistemological investigations into many areas of
mathematics.
Discarding the burden of finding a foundation for all mathematics allows
the local formalization of specific topics in a manner faithful to the
individual discipline. Thus algebraic geometry can be seen as the
study of relations on (algebraically closed) fields defined by conjunctions
of equations – the essentials of
Weil-style algebraic geometry are captured directly by such a formalization.
This sort of example supports the first aspect of Thesis1. Key to the
second aspect is the identification (foreshadowed by, for example, Manders
[Man87]) of clarity as an equally important facet of epistemology as
reliability. We contrast ‘quantifier elimination of T’ (all first order
definable relations on models of a theory have very simple definitions) and
structure theorems for models of T as two ways to clarify our understanding
of the area of mathematics formalized in T.

Further we place Shelah’s ‘method of dividing lines’ as a new development
in the age-old study of classification.
Many of the sections have a dual purpose. The both shine a new light on a
traditional problem and build understanding of the overall project. For
example, Chapter 5.1, From analogy to theorem to method,
builds on work of Corfield, Grosholz, and Schlimm explaining how analogies
between logic and algebra led to the topological notion of the Stone space
(the set of maximal consistent sets of formulas). It shows
that the semantical interpretation of types became a fundamental to
construct models. In Chapter 7.2, the combinatorial tools to construct
infinite sets of indiscernibles are introduced by a discussion of the
Leibnitz principle of the ‘identity of indiscernibles’, even though it is
clearly false from the model theoretic viewpoint.

For Thesis 3, we demonstrate that formalization in first order logic gives
finer information about such long standing topics in mathematical
philosophy as categoricity, definability and axiomatizability. In a
novel analysis we demonstrate the advantages in terms of Meadows criteria
[Mea13] of a categorical L ω 1 ,ω - axiomatization of the natural numbers
over the usual second order one.

In response to a question of Detlefsen, we define a property (of a theory)
to be virtuous if it has significant mathematical consequences for the
theory or its models. Then we show that categoricity of a second order
sentence lacks virtue, L ω 1 ,ω -
categoricity has some virtues but categoricity in power of a first order
theory is most virtuous  because of its structural consequences

Thesis 4 reflects the fact that geometry, in at least three avatars,
pervades the book. Euclidean geometry is the major example for the
discussion of the role of axioms, leading to a new axiomatization of the
geometry of polygons and circles.

 Mid-twentieth century algebraic geometry in the guise of the theory of
algebraically closed fields is a prototype for ℵ 1 -categorical theories;
generalizing this notion leads to a structure theory for models of a wide
but sharply-defined class of theories. The chief technical tool for this
structure theory is the notion of a combinatorial geometry (matroid). This
notion aims at explicating the intuitive notion of ‘dimension’. Finally,
the entanglement of model theory with classical mathematics is demonstrated
by Hrushovski’s theorem defining one of the classical groups (standard
structures that play an important role in many areas of mathematics) in any
model satisfying very abstract stability theoretic conditions, which at
first sight have no algebraic content.


The book presents the notion of formalization as a powerful tool. But this
tool is employed in the context of viewing mathematics as an autonomous
entity. By focusing on a particular area of mathematics, one can  see
successive presentations of the subject as exposing the topic with
increasing clarity. The demand for formalization both drives the
development of more precise formulations and in the case of first order
logic, allows via the stability theory, the rigorous comparison of various
areas.

Please excuse any delay in replying to comments. I will traveling the next
couple of days.

John T. Baldwin
Professor Emeritus
Department of Mathematics, Statistics,
and Computer Science M/C 249
jbaldwin at uic.edu
851 S. Morgan
Chicago IL
60607
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