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The historical and epistemological reflection on the applications
of mathematical techniques to the Sciences of Nature - physics,
biology, chemistry, and geology - today generates attention and
interest because of the increasing use of mathematical models in
all sciences and their high level of sophistication. The goal of
the meeting and the papers collected in this proceedings volume is
to give physicists, biologists, mathematicians, and historians of
science the opportunity to share information on their work and
reflect on the and mathematical models are used in the natural
sciences today and in way mathematics the past. The program of the
workshop combines the experience of those working on current
scientific research in many different fields with the historical
analysis of previous results. We hope that some novel
interdisciplinary, philosophical, and epistemological
considerations will follow from the two aspects of the workshop,
the historical and the scientific. This proceedings includes papers
presented at the meeting and some of the results of the discussions
that took place during the workshop. We wish to express our
gratitude to Sergio Monteiro for all his work, which has been
essential for the successful publication of these proceedings. We
also want to thank the editors of Kluwer AcademidPlenum Publishers
for their patience and constant help, and in particular Beth Kuhne
and Roberta Klarreich. Our thanks to the fallowing institutions:
-Amministrazione Comunale di Arcidosso -Comunita Montana del Monte
Amiata .Center for the History of Physics, UCLA -Centre F."
Determinism, holism and complexity: three epistemological attitudes
that have easily identifiable historical origins and developments.
Galileo believed that it was necessary to "prune the impediments"
to extract the mathematical essence of physical phenomena, to
identify the math ematical structures representing the underlying
laws. This Galilean method was the key element in the development
of Physics, with its extraordinary successes. Nevertheless the
method was later criticized because it led to a view of nature as
essentially "simple and orderly," and thus by choosing not to
investigate several charac teristics considered as an "impediment,"
several essential aspects of the phenomenon under investigation
might be left out. The Galilean point of view also contains an
acknowledgement of the central role played by the causal nexus
among phenomena. The mechanistic-deterministic de scription of
reality - for instance, a la Laplace - although acknowledging that
it is not possible to predict phenomena exactly owing to unavoid
able measurement error, is based on the recognition of the their
causal nature, even in an ontological sense. Consequently,
deterministic predic tion became the methodological fulcrum of
mathematical physics. But although mechanistic determinism has had
and, in many cases, still has, considerable success in Physics, in
other branches of science this situa tion is much less favourable."
By modern analytic mechanics we mean the classical mechanics of
today, that is, the mechanics that has proven particularly useful
in understanding the universe as we experience it from the solar
system, to particle accelerators, to rocket motion. The
mathematical and numerical techniques that are part of this
mechanics that we present are those that we have found to be
particularly productive in our work in the subject. The balance of
topics in this book is somewhat different from previous texts. We
emphasize the use of phase space to describe the dynamics of a
system and to have a qualitative understanding of nonlinear
systems. We incorporate exercises that are to be done using a
computer to solve linear and nonlinear problems and to have a
graphical representation of the results. While analytic solutions
of physics problems are to be prefer. red, it is not always
possible to find them for all problems. When that happens,
techniques other than analysis must be brought to bear on the
problem. In many cases numerical treatments are useful in
generating solutions, and with these solutions often come new
insights. These insights can sometimes be used for making further
analytic progress, and often the process is iterative. Thus the
ability to use a computer to solve problems is one of the tools of
the modern physicist. Just as analytic problem-solving enhances the
student's understanding of physics, so will using the computer
enhance his or her appreciation of the subject.
The historical and epistemological reflection on the applications
of mathematical techniques to the Sciences of Nature - physics,
biology, chemistry, and geology - today generates attention and
interest because of the increasing use of mathematical models in
all sciences and their high level of sophistication. The goal of
the meeting and the papers collected in this proceedings volume is
to give physicists, biologists, mathematicians, and historians of
science the opportunity to share information on their work and
reflect on the and mathematical models are used in the natural
sciences today and in way mathematics the past. The program of the
workshop combines the experience of those working on current
scientific research in many different fields with the historical
analysis of previous results. We hope that some novel
interdisciplinary, philosophical, and epistemological
considerations will follow from the two aspects of the workshop,
the historical and the scientific* This proceedings includes papers
presented at the meeting and some of the results of the discussions
that took place during the workshop. We wish to express our
gratitude to Sergio Monteiro for all his work, which has been
essential for the successful publication of these proceedings. We
also want to thank the editors of Kluwer AcademidPlenum Publishers
for their patience and constant help, and in particular Beth Kuhne
and Roberta Klarreich. Our thanks to the fallowing institutions:
-Amministrazione Comunale di Arcidosso -Comunita Montana del Monte
Amiata *Center for the History of Physics, UCLA -Centre F.
By modern analytic mechanics we mean the classical mechanics of
today, that is, the mechanics that has proven particularly useful
in understanding the universe as we experience it from the solar
system, to particle accelerators, to rocket motion. The
mathematical and numerical techniques that are part of this
mechanics that we present are those that we have found to be
particularly productive in our work in the subject. The balance of
topics in this book is somewhat different from previous texts. We
emphasize the use of phase space to describe the dynamics of a
system and to have a qualitative understanding of nonlinear
systems. We incorporate exercises that are to be done using a
computer to solve linear and nonlinear problems and to have a
graphical representation of the results. While analytic solutions
of physics problems are to be prefer. red, it is not always
possible to find them for all problems. When that happens,
techniques other than analysis must be brought to bear on the
problem. In many cases numerical treatments are useful in
generating solutions, and with these solutions often come new
insights. These insights can sometimes be used for making further
analytic progress, and often the process is iterative. Thus the
ability to use a computer to solve problems is one of the tools of
the modern physicist. Just as analytic problem-solving enhances the
student's understanding of physics, so will using the computer
enhance his or her appreciation of the subject.
Determinism, holism and complexity: three epistemological attitudes
that have easily identifiable historical origins and developments.
Galileo believed that it was necessary to "prune the impediments"
to extract the mathematical essence of physical phenomena, to
identify the math ematical structures representing the underlying
laws. This Galilean method was the key element in the development
of Physics, with its extraordinary successes. Nevertheless the
method was later criticized because it led to a view of nature as
essentially "simple and orderly," and thus by choosing not to
investigate several charac teristics considered as an "impediment,"
several essential aspects of the phenomenon under investigation
might be left out. The Galilean point of view also contains an
acknowledgement of the central role played by the causal nexus
among phenomena. The mechanistic-deterministic de scription of
reality - for instance, a la Laplace - although acknowledging that
it is not possible to predict phenomena exactly owing to unavoid
able measurement error, is based on the recognition of the their
causal nature, even in an ontological sense. Consequently,
deterministic predic tion became the methodological fulcrum of
mathematical physics. But although mechanistic determinism has had
and, in many cases, still has, considerable success in Physics, in
other branches of science this situa tion is much less favourable."
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