Table Of ContentThe
Pythagorean
World
Why Mathematics is
Unreasonably Effective in Physics
Jane McDonnell
The Pythagorean World
Jane McDonnell
The Pythagorean
World
Why Mathematics Is Unreasonably Effective In
Physics
Jane McDonnell
Philosophy Department Clayton,Victoria
Australia
ISBN 978-3-319-40975-7 ISBN 978-3-319-40976-4 (eBook)
DOI 10.1007/978-3-319-40976-4
Library of Congress Control Number: 2016957514
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Prefa ce
Don’t be surprised then, Socrates, if it turns out repeatedly that we won’t
be able to produce accounts on a great many subjects—on gods or the
coming to be of the universe—that are completely and perfectly consistent
and accurate. Instead, if we can come up with accounts no less likely than
any, we ought to be content, keeping in mind that both I, the speaker, and
you, the judges, are only human.
Plato, T imaeus 29c
v
Acknowledgements
Many people helped with the preparation of this book. Th e following
deserve special recognition.
Th anks to Graham Oppy for invaluable discussions, comments, sug-
gestions and direction. Th anks to Monima Chadha for her encour-
agement. Th anks to the students and staff of the Monash University
Philosophy Department for general support and assistance. Th anks to
Robert Griffi ths for patiently answering my questions about consistent
histories quantum mechanics. Th anks to the examiners of my thesis, Eric
Steinhart and Peter Forrest, for their suggested improvements.
M y special thanks go to my husband, Michael McDonnell, for his
critical insights. Without his support, this book would not have been
possible.
Any and all mistakes or omissions are mine alone.
vii
Abstract In this book, I argue that many problems in the philosophy of
science and mathematics (in particular, the unreasonable eff ectiveness of
mathematics in physics) can only be addressed within a broader meta-
physical framework which provides a coherent world view. I attempt to
develop such a framework and draw out its consequences. Th e attempt is
in two parts: fi rstly, I develop a speculative framework based on an anal-
ogy to set theory, then I combine elements of the framework with ideas
from Leibnizian monadology and consistent histories quantum theory to
introduce (what I call) quantum monadology. Th e two parts focus on dif-
ferent aspects of the problem and should be viewed as stages on the way
to a fi nal formulation. Th e inspiration for the book came from Plato’s
Timaeus and Wigner’s comments on quantum mechanics. As it turned
out, Leibniz’s M onadology became a third key source.
Contents
1 Introduction 1
2 Th e Applicability of Mathematics 3
3 Th e Role of Mathematics in Fundamental Physics 69
4 One True Mathematics 125
5 What Mathematics Is About 179
6 Actuality from Potentiality 223
7 Conclusion 299
Appendix 1 331
ix
x Contents
Appendix 2 349
References 361
Index 383
1
Introduction
In his article ‘Th e Unreasonable Eff ectiveness of Mathematics in the
Natural Sciences’, the physicist Eugene Wigner asks “What is mathemat-
ics?”, “What is physics?” and “Why is mathematics so unreasonably eff ec-
tive in physics?”. Th ese are the key questions addressed in this book.
Depending on which philosophical school one adheres to, mathematics
might be a form of logic (logicism), a construction of the human mind (intu-
itionism), a game played with symbols (formalism), a language describing
the properties of real, abstract entities (platonism) or a language describing
fi ctional entities (fi ctionalism). One thing which all schools have to explain
is the eff ectiveness of mathematics in physics. It is a striking feature of math-
ematics that it allows us to model and predict the behaviour of physical sys-
tems to an amazing degree of accuracy. For example, the predicted magnetic
moment of the electron agrees with the current experimental value to an
accuracy of one part in a trillion.
O ne of the oldest explanations of the eff ectiveness of mathematics
in physics is that, in some profound way, the structure of the world is
mathematical. Th e Pythagoreans believed that “everything is number”.
If we interpret this explanation as saying that mathematical structure is
© Th e Author(s) 2017 1
J. McDonnell, Th e Pythagorean World,
DOI 10.1007/978-3-319-40976-4_1
