- Nico Bertoloni Meli, "Beyond inertia: Rethinking Mechanics and
Motion in the 17th Century"
Over the last few decades studies on seventeenth-century mechanics and
motion have focused on a range of themes, the single most prominent one
being the study of laws of nature. In my work I adopt a different
perspective centered on the objects of investigation, such as the inclined
plane, the pendulum, the string, and colliding bodies, for example. I argue
that this perspective allows us to reconsider in a fresh light issues such
as the role of experiments and the mathematization of nature. My focus on
objects does not lead to an exclusion of laws, however, but helps us to
reconsider their origin and modes of operation.
- Mary Terrall, "Vis viva Revisited"
The vis viva controversy is a canonical site for analysing irreconcilable
philosophical positions in early-modern physics. Viewed in light of
subsequent formulations of the concepts of energy and momentum, the endless
wrangling of the participants reduces to confusion, and not much else. What
more can be said about this controversy, disdainfully characterized in 1743
by d'Alembert (among others) as "a dispute of words too undignified to
occupy philosophers any longer"? This paper looks at the trajectory of
the dispute from the point of view of the alliances and antipathies
connecting and dividing the antagonists, particularly in the period from the
1720s to the early 1740s. These relations were governed by philosophical
commitments, to be sure, but also by competing mathematical techniques.
Tensions in institutional and personal allegiances were played out in
academic prize essays, in the periodical press and in books. The controversy
simmered for many years within the small international community of
mathematicians, flaring up occasionally into more or less vitriolic
arguments that were never resolved. It bubbled over into greater public
visibility when Emilie Du Chtelet and Voltaire entered the fray in the
1740s. By this time, the arcana of 17th-century dynamics had become the
stuff of Enlightenment posturing. Why did Voltaire care about this question?
Why did he think his public would care? And why were others, arguably more
mathematically astute, focusing on other physical laws and concepts
altogether? Inspired by these questions, the paper investigates the
interests and motivations at play at several key moments in the complex
history of this dispute.
- Michael Friedman, "Physics, Philosophy, and the Foundations of
Geometry"
The development of Einstein's theory of relativity against the background of
late nineteenth century work on the foundations of geometry provides an
unsurpassed example of the multiple levels of interaction between physics,
mathematics, and philosophical reflection on the two. On the one hand,
Einstein's creation of the general theory of relativity was essentially
mediated by his own philosophical engagement with earlier reflections on the
foundations of geometry by Helmholtz and Poincare (which reflection was
itself intimately related, of course, to the nineteenth century development
of non-Euclidean geometries). On the other hand, this same interrelated
series of developments, culminating in Einstein's work, was taken by logical
empiricist philosophers to support a characteristically twentieth century
version of geometrical conventionalism. This philosophical conclusion
ultimately proved to be incoherent, but seeing this in detail reveals the
incredible richness of mutual interaction between physics, mathematics, and
philosophy that actually resulted in Einstein's theory.
- Mark Wilson, "A Funny Thing Happened on the Way to the
Formalism"
Hilbert's sixth problem requests an axiomatic formulation of classical
mechanics, with special attention to the question of how competing
foundational ideas might be reconciled. In the context of its time, this
constituted a quite reasonable request but how should we look at these
matters today? It seems to me there are clear reasons why a formalism of the
sort desired can't be produced and the salient considerations offer
interesting morals for the philosophy of language. I'll look at this
question in the context of Duhem's complaints about the practices of
physicists such as Lord Kelvin.
- Brad Bassler, The Parafinite and the Natural World
I look at a tradition of attempts, beginning with Galileo, Descartes and
Leibniz, to develop a notion of the mathematically indefinitely large, which
I refer to as the parafinite. The development of the parafinite, whether
understood as neither finite nor infinite (Galileo), as closely associated
with the infinite (Descartes), or as infinite (Leibniz), emerged in
conjunction with a need to find mathematical descriptions of various aspects
of the natural world. Two issues are deeply interwoven: the mathematical
need for resolutive adequacy, i.e. the requirement that mathematical tools
be sufficiently powerful to solve mathematical problems, and the need for
descriptive adequacy, in particular for the description of the natural
world. In a contemporary setting, the investigation of the parafinite has
been reinitiated, in part, in attempts to provide contemporary
characterizations of the Leibnizian parafinite. I consider two: the
description of Leibnizian infinitesimals by Robinson and Lakatos in terms of
non-standard analysis, and by Shaughan Lavine in terms of Mycielski's
"analysis without actual infinity." The latter approach is more
fruitful than the former, not least because it opens new possibilities for
addressing questions about the relation between mathematical description and
the world it describes, which I briefly illustrate in the context of recent
work in chaotic dynamics.
- Jordi Cat, "Representation as Theory: How and Why Mathematical
Representation Makes a Difference in Physical Theory"
It is typically argued, or taken for granted, that the meaning of physical
theories is wholly contained in their physical formalism or collection of
models. Purely mathematical formalism is relegated to the status of an
uninterpreted formal calculus whose different versions are merely different
linguistic formulations of an essentially non-linguistic physical theory,
itself centered around dynamical equations. In this paper, I explore the
notion that in mathematical formalism itself we can distinguish purely
symbolic from non-symbolic content; this distinction has important
consequences for our understanding of physical theory and phenomena. Between
empty formal computational rules and dynamical equations, there exists a
range of hybrids, namely, constitutive abstract physical models.
These models, considered as mediating representations and not as mere formal
structure, can shape, guide, and enrich the story that theories tell about
phenomena. I list a number of examples and focus on some related to the
history of field theories, for example: the application of synthetic
quaternionic and vector representations of equations alongside Cartesian
representation of components, the application of differential equations as
emblematic of contiguous action, the dynamical interpretation of
differential operators, and the Eulerian and the Lagrangian representations
of hydrodynamics. Finally, I suggest that being mindful of these kinds of
representations and their variability may provide a powerful heuristic in
the exploration and development of future physics.
- John Norton, "Einstein and the Canon of Mathematical
Simplicity"
Einstein proclaimed that we could discover true laws of nature by seeking
those with the simplest mathematical formulation. He came to this viewpoint
later in his life. In his early years and work he was quite hostile to this
idea. Einstein did not develop his later Platonism from a priori reasoning
or aesthetic considerations. He learned the canon of mathematical simplicity
from his own experiences in the discovery of new theories, most importantly,
his discovery of general relativity. Through his neglect of the canon, he
realized that he delayed the completion of general relativity by three years
and nearly lost priority in discovery of its gravitational field equations.
- Andrew Warwick, "University education and the history of
mathematical physics in the longue duree"
From its beginnings in the middle decades of the seventeenth century, the
new discipline of 'physico-mathematics' rapidly became extremely technical.
Although most of its practitioners were university educated, the discipline
itself did not fit easily into a university curriculum intended to provide a
general education, and based mainly on oral techniques of teaching and
examining. In this paper I shall explore how mathematical physicists (from
'physico-mathematicians' to 'theoretical physicists') learned their craft,
and how they passed on their knowledge from one generation to another.
Taking Cambridge University as my example, I shall pay special attention to
the decades around 1800, when the mathematician's esoteric knowledge became
central to undergraduate studies. How did this transformation alter the
practice of the mathematical sciences and what can we learn from it about
the nature of mathematical knowledge?
- Michael Dickson, "Bohr's Attitude Towards Mathematical
Formalism"
As early as 1915 Niels Bohr attempted to provide a "formally consistent
theory", or "a general foundation", of quantum mechanics, as
it was known at the time. Four more or less physically motivated 'axioms'
(Bohr calls them 'assumptions') formed the heart of his proposed foundation.
Bohr quickly realized that they were too limited in scope, and of course
quantum theory had to wait until 1925 (or 1932, depending on how you count)
for a true theoretical unification, in the form of Heisenberg's matrix
mechanics and Schrödinger's wave mechanics (or as they were themselves
unified in von Neumann's Hilbert space mechanics).
Bohr is often dealt quite a harsh blow for not appreciating the mathematical
work of Heisenberg, Schrödinger, Dirac, Jordan, and others who forged the
mathematical consolidation of the old quantum theory, and the contrast in
character between his 'physical' axioms of 1915 and their 'mathematical'
axioms of the 1920s might be construed as evidence that Bohr was after
something quite different from what they produced. Indeed he was. However,
this fact alone does not entail that Bohr failed to appreciate the
importance of the mathematization of quantum theory that occurred in the
1920s. I will argue that in fact Bohr absorbed this mathematization into his
own view of quantum theory, and developed a fairly sophisticated view of the
relationship between mathematical formalism and physical theory as a result.
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