Table Of ContentNaturalising Badiou
Naturalising Badiou
Mathematical Ontology and
Structural Realism
Fabio Gironi
IRC Postdoctoral Research Fellow in Philosophy,
University College Dublin, Ireland
© Fabio Gironi 2015
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First published 2015 by
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To Jasper, my companion at the bonfire
Contents
Acknowledgements i x
Note about Citations xii
Introduction 1
1 Badiou’s Mathematical Ontology 9
1.1 Death(s) of God: for axiomatic immanence 9
1.2 Subtractive ontology 12
1.3 Set theory 21
1.4 The Void 27
1.5 Secularising infinity 29
2 The Ontological and the Empirical: Naturalist Objections 3 4
2.1 Pythagoreanism and materialism 36
2.2 Philosophy of mathematics to the rescue 4 1
2.3 The applicability of mathematics 4 5
2.4 Naturalist explanations of mathematics 4 9
2.5 Two naturalising strategies 57
2.6 The matherialist option 60
3 Taking a Stance on Realism and Naturalism 63
3.1 Scientific realism 6 4
3.2 The realist stance 6 7
3.3 Transgressive naturalism 72
3.4 Historicist excursus: Collingwood and Bachelard 7 5
3.5 The realist’s task 80
4 Structural Realisms 86
4.1 Scientific realism, its adbuctive defence ...
and its discontents 86
4.2 Epistemic structural realism 90
4.3 Ontic structural realism 93
4.4 Mathematical structuralism 102
4.5 Structures of what? 108
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viii Contents
5 Truth and Randomness 112
5.1 Against fundamental ontology 1 12
5.2 Idealism and rationality 119
5.3 Truth as abductive process 124
5.4 Metamathematics and randomness 1 35
Conclusion 1 43
Notes 149
Bibliography 2 10
Index 2 31
Acknowledgements
I would like to enumerate a long and impressive list of fellow students
and prominent scholars to thank for the priceless insights I have drawn
from long conversations with them, over late-night glasses of single malt
and endless walks in the misty countryside. However, this work is mostly
the product of solitary struggle, mainly involving sparkling water and a
comfy armchair, and such a list is thus very short indeed. Not being the
most conversational type, my most productive work is done in isolation,
with the sole silent, yet not mute, company of books.
Of the few individuals who have been instrumental in the successful
completion of this book my gratitude goes, first of all, to Chris Norris.
As one of the handful of scholars worldwide capable of following and
assessing my conceptual journeys across the lands of analytic and conti-
nental philosophy, his guidance has always been attentive but unob-
trusive. He has time and again frustrated my hopes to try and refer to
a book he hadn’t already read, he has voiced his disagreements with
some of my ideas as a peer, without ever pressuring me to recant them,
and he often seemed to have more faith in my project than I ever did.
The breadth of his interests and knowledge, coupled with his humble
and unassuming attitude towards everyone, has been an outstanding
example of scholarly virtue to me. I have been immensely privileged to
be his student.
For good or bad, my intellectual trajectory was radically modified
by the discovery, sometime in early 2009, of a highly active internet
community of philosophers that was flourishing around the then-novel
movement of ‘speculative realism’. Whatever the intrinsic merits of this
dubiously unitary movement, the virtual encounter with many bright
young thinkers loosely associated with it has been tremendously impor-
tant for the development of my thought. It forced me continuously
to try to catch-up with brighter minds, a chase (still in process) which
time and again shattered my fragile self-esteem, and punctuated by the
occasional, dreaded gaze into the gaping abyss where The Books You
Haven’t Read Yet lurk. In one way or another several of these dozens of
virtual acquaintances have influenced me. Inevitably, however, some of
these had a greater impact on my philosophical commitments, a set that
luckily comes to largely intersect with those I now have the privilege to
call friends. Pete Wolfendale has for me variously taken on the garb of
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