Valérie Lynn Therrien BA

September 1, 2023 until February 27, 2024

Affiliation: McGill University

Research title for a study about:

Models of Mathematics

My current research centers on the question of how to adjudicate an appropriate
background logic for axiomatizations of mathematics. Currently,
first-order logics are the default background logic. They have the advantage
of being simple yet expressive, deductively complete, but has two main disadvantages.
The first is that it is not categorical, which means that there
are infinitely many non-isomorphic models for e.g., the natural number sequence
0, 1, 2, 3, 4, . . . — most including strange entities like ‘non-standard’
numbers capable of being greater than all natural numbers, and in which
addition and multiplication aren’t computable! The second disadvantage is
that making first-order logics expressive enough to axiomatize the natural
numbers renders it incomplete! In many ways, first-order logics simply are
not up to the task. Second-order logics are deductively incomplete, but have
the advantage of being categorical, which makes them far more suitable to
talk with precision about a mathematical structure. My hypothesis for the
curious side-lining of second-order logics has precisely to do with the fascinating
panoply of non-standard models, which required the development of
a newfield of mathematics to investigate : model theory. I intend to generate
a history of model theory and use this history as a case study for Philosophy
of Mathematical Practice in order to improve our understanding of scientific
progress and decision-making with respect to our background theories
when the advantages and disadvantages aren’t themselves decisive.

Lecture

Towards a History of Model Theory

Philosophy of Science Colloquium

Date: 2023 January 18, postponed to February 22, 2024

Time: 3-4.30 pm CET

Venue: NIG, Universitätsstraße 7, 1010 Wien, SR 2H

Abstract:

My current research centers on the question of how to adjudicate an appropriate
background logic for axiomatizations of mathematics. Currently,
first-order logics are the default background logic. They have the advantage
of being simple yet expressive, deductively complete, but has two main disadvantages.
The first is that it is not categorical, which means that there
are infinitely many non-isomorphic models for e.g., the natural number sequence
0, 1, 2, 3, 4, . . . — most including strange entities like “non-standard”
numbers capable of being greater than all natural numbers, and in which
addition and multiplication aren’t computable! The second disadvantage is
that making first-order logics expressive enough to axiomatize the natural
numbers renders it incomplete! In many ways, first-order logics simply are
not up to the task. Second-order logics are deductively incomplete, but have
the advantage of being categorical, which makes them far more suitable
to talk with precision about a mathematical structure. As such, I intend to
use the history of model theory as of scientific progress and decision-making
with respect to our background theories when the advantages and disadvantages
aren’t themselves decisive. But first, this history must be generated,
and the early history of model theory is what I will focus on for this talk.

Report

During my six month fellowship at the IVC, my main accomplishments were :
1) completing a full draft of the pre-history of model theory (algebraic logic) from the English algebraists to Schröder (a tim; and 2) completing intensive German lesson. Being based in Canada, I was also able to grow my European network substantially, giving three invited talks in Vienna and Budapest and participating in all sessions of the Institute’s Logic Café and Philosophy of Science Colloquium, as well as two reading groups. The opportunity provided to me by the IVC was invaluable, as well as the opportunity to work with Georg Schiemer. I was well-surrounded by colleagues both in my specific field (logic and history and philosophy of mathematics) as well as in broader philosophy of science. The environment at the IVC’s centrally located office fosters both working tightly with like-minded colleagues and small-group interchanges and discussions of idea both within the Institute and the connected Philosophy Department a mere block away. As I complete my dissertation, my stay with the IVC has been formative and provided me with exceptional practical resources to prepare myself for the upcoming job market.