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# Fermat reals [Elektronische Ressource] : nilpotent infinitesimals and infinite dimensional spaces / vorgelegt von Paolo Giordano

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Fermat RealsNilpotent In nitesimals and In nite DimensionalSpacesDissertationzurErlangung des Doktorgrades (Dr. rer. nat.)der Mathematisch-Naturwissenschaftlichen FakultatderRheinischen Friedrich-Wilhelms-Universitat Bonnvorgelegt vonGIORDANOPAOLOausMonzaBonn (2009)Angefertigt mit Genehmigung der Mathematisch-NaturwissenschaftlichenFakultat der Rheinischen Friedrich-Wilhelms-Universitat Bonn1. Gutachter: Prof. Dr. D.h.c. Sergio Albeverio2. Gutachter: Prof. Dr. Peter KoepkeTag der Promotion: 21 December 2009Erscheinungsjahr: 2010ContentsI Algebraic and order properties of Fermat reals 11 Introduction and general problem 31.1 Motivations for the name \Fermat reals" . . 72 De nition of Fermat reals 92.1 The basic idea . . . . . . . . . . . . . . . . . 92.2 First properties of little-oh polynomials . . . 122.3 Equality and decomposition of Fermat reals 132.4 The ideals D . . . . . . . . . . . . . . . . . 18k2.5 Products of powers of nilpotent in nitesimals 212.6 Identity principle for polynomials . . . . . . 242.7 Invertible Fermat reals . . . . . . . . . . . . 262.8 The derivation formula . . . . . . . . . . . . 283 Equality up to k-th order in nitesimals 333.1 Introduction . . . . . . . . . . . . . . . . . . 333.2 Equality up to k-th order in nitesimals . . . 363.3 Cancellation laws up to k-th order in nites-imals . . . . . . . . . . . . . . . . . . . . . . 423.4 Applications to Taylor’s formulae . . . . . . 473.

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##### Mathematics

Informations

Fermat Reals
Nilpotent In nitesimals and In nite Dimensional
Spaces
Dissertation
zur
Erlangung des Doktorgrades (Dr. rer. nat.)
der Mathematisch-Naturwissenschaftlichen Fakultat
der
Rheinischen Friedrich-Wilhelms-Universitat Bonn
vorgelegt von
GIORDANOPAOLO
aus
Monza
Bonn (2009)Angefertigt mit Genehmigung der Mathematisch-Naturwissenschaftlichen
Fakultat der Rheinischen Friedrich-Wilhelms-Universitat Bonn
1. Gutachter: Prof. Dr. D.h.c. Sergio Albeverio
2. Gutachter: Prof. Dr. Peter Koepke
Tag der Promotion: 21 December 2009
Erscheinungsjahr: 2010Contents
I Algebraic and order properties of Fermat reals 1
1 Introduction and general problem 3
1.1 Motivations for the name \Fermat reals" . . 7
2 De nition of Fermat reals 9
2.1 The basic idea . . . . . . . . . . . . . . . . . 9
2.2 First properties of little-oh polynomials . . . 12
2.3 Equality and decomposition of Fermat reals 13
2.4 The ideals D . . . . . . . . . . . . . . . . . 18k
2.5 Products of powers of nilpotent in nitesimals 21
2.6 Identity principle for polynomials . . . . . . 24
2.7 Invertible Fermat reals . . . . . . . . . . . . 26
2.8 The derivation formula . . . . . . . . . . . . 28
3 Equality up to k-th order in nitesimals 33
3.1 Introduction . . . . . . . . . . . . . . . . . . 33
3.2 Equality up to k-th order in nitesimals . . . 36
3.3 Cancellation laws up to k-th order in nites-
imals . . . . . . . . . . . . . . . . . . . . . . 42
3.4 Applications to Taylor’s formulae . . . . . . 47
3.5 Extension of some results to D . . . . . . . 541
3.6 Some elementary examples . . . . . . . . . . 56
4 Order relation 61
4.1 In nitesimals and order properties . . . . . . 61
iCONTENTS
4.2 Order relation . . . . . . . . . . . . . . . . . 64
4.2.1 Absolute value . . . . . . . . . . . . 72
4.3 Powers and logarithms . . . . . . . . . . . . 73
4.4 Geometrical representation of Fermat reals . 74
II In nite dimensional spaces 79
5 Approaches to1-dimensional di . geom. 81
5.1 Approaches to in nite-dimensional spaces . . 81
5.2 Banach manifolds and locally convex vector spaces . 85
5.3 The convenient vector spaces settings . . . . 88
5.4 Di eological spaces . . . . . . . . . . . . . . 91
5.5 Synthetic di erential geometry . . . . . . . . 96
6 The cartesian closure of a category of gures105
6.1 Motivations and basic hypotheses . . . . . . 105
6.2 The cartesian closure and its rst properties 107
6.3 Categorical properties of the cartesian closure 111
n7 The categoryC 115
n7.1 Observables onC spaces and separated spaces115
n7.2 Manifolds as objects ofC . . . . . . . . . . 118
n7.3 Examples ofC spaces and functions . . . . 119
8 Extending smooth spaces with in nitesimals123
8.1 Introduction . . . . . . . . . . . . . . . . . . 123
8.1.1 Nilpotent paths . . . . . . . . . . . . 124
18.1.2 Little-oh polynomials inC . . . . . 127
8.2 The Fermat extension of spaces and functions 132
8.3 The category of Fermat spaces . . . . . . . . 133
9 The Fermat functor 141
9.1 Putting a structure on the sets X . . . . . 141
iiCONTENTS
9.2 The Fermat functor preserves product of man-
ifolds . . . . . . . . . . . . . . . . . . . . . . 142
9.2.1 Figures of Fermat spaces . . . . . . . 145
19.3 The embedding of manifolds in C . . . . . 147
9.4 The standard part functor cannot exist . . . 148
9.4.1 Smooth functions with standard values150
10 Logical properties of the Fermat functor 155
10.1 Basic logical properties of the Fermat functor 155
10.2 The general transfer theorem . . . . . . . . . 166
III The beginning of a new theory 177
11 Calculus on open domains 179
11.1 Introduction . . . . . . . . . . . . . . . . . . 179
11.2 The Fermat-Reyes method . . . . . . . . . . 180
11.3 Integral calculus . . . . . . . . . . . . . . . . 197
12 Calculus on in nitesimal domains 209
12.1 The generalized Taylor’s formula . . . . . . 209
12.2 Smoothness of derivatives . . . . . . . . . . 219
13 In nitesimal di erential geometry 229
13.1 Tangent spaces and vector elds . . . . . . . 229
13.2 In nitesimal integral curves . . . . . . . . . 240
13.3 Ideas for the calculus of variations . . . . . . 244
14 Further developments 251
14.1 First order in nitesimals whose product is
not zero . . . . . . . . . . . . . . . . . . . . 251
14.2 Relationships with Topos theory . . . . . . . 253
14.3 A transfer theorem for sentences . . . . . . . 254
iiiCONTENTS
14.4 Two general theorems for two very used tech-
niques . . . . . . . . . . . . . . . . . . . . . 254
14.5 In nitesimal di erential geometry . . . . . . 255
14.6 Automatic di erentiation . . . . . . . . . . 256
14.7 Calculus of variations . . . . . . . . . . . . . 256
14.8 In nitesimal calculus with distributions . . . 256
14.9 Stochastic in nitesimals . . . . . . . . . . . 257
14.10In nite numbers and nilpotent in nitesimal 258
IV Appendices 261
A Some notions of category theory 263
A.1 Categories . . . . . . . . . . . . . . . . . . . 263
A.2 Functors . . . . . . . . . . . . . . . . . . . . 267
A.3 Limits and colimits . . . . . . . . . . . . . . 269
A.4 The Yoneda embedding . . . . . . . . . . . . 272
A.5 Universal arrows and adjoints . . . . . . . . 273
B Other theories of in nitesimals 277
B.1 Nonstandard Analysis . . . . . . . . . . . . 278
B.2 Synthetic di erential geometry . . . . . . . . 285
B.3 Weil functors . . . . . . . . . . . . . . . . . 287
B.4 Surreal numbers . . . . . . . . . . . . . . . . 289
B.5 Levi-Civita eld . . . . . . . . . . . . . . . . 297
Bibliography 308
ivList of Figures
4.1 How to guess that hk = 0 for two rst order in nitesimals h, k2D 62
4.2 The function representing the Fermat real dt 2D . . . . . . . . 752 3
4.3 Some rst order in nitesimals . . . . . . . . . . . . . . . . . . . 77
4.4 The product of two . . . . . . . . . . . . . . . . . . 77
4.5 Some higher order in nitesimals . . . . . . . . . . . . . . . . . 78
4.6 Di erent cases in which x <y . . . . . . . . . . . . . . . . . 78i i
11.1 Intervals for the recursive de nition of a primitive . . . . . . 201
13.1 An example of space which is not inf-linear at m2X. . . . . 235
13.2 Explanation of the de nition of integral curve . . . . . . . . . 241
13.3 An example of function(t;h; ; ; ) fort = 0:5,t = 1,1 2 1
t +h = 1:5, t +h + = 2:5. . . . . . . . . . . . . . . . . . . . 2492
vAcknowledgments
It is not easy to nd oneself with a so called unusual curriculum vitae.
After one wins a Marie Curie fellowship, without possessing a PhD, and
after the funding for several research projects, the European Commission
considered you as a senior researcher. Anyway, one frequently has to listen
phrases of the type \but you haven’t got. . . ", \but you. . . ", i.e.: but you are
not in the mainstream system as we expected you to be. Therefore, my best
thanks go to all the people that show me the courage to believe in someone
that is not in the mainstream system.
My wife Cristina, my parents, my daughter Sara, who has still the free-
dom to be out of any mainstream system.
In the same group, I would also like to thank Prof. S. Albeverio, for his
great encouragement and huge patience.
Finally, the author would also like to thank Pietro Castelluccia for his
suggestions about the non easy problem of the existence of the standard
part functor.
If we do not believe in the existence of God, then, from Godel’s onto-
logical theorem it follows that an absolute, necessary and not only possible
moral system cannot exists. But I believe that the human kind can achieve
this type of moral system, so I have to believe in God.
If you further assume, suitably formalized by a rigorous mathematical
language, that any good thing has in God its rst cause and that mathe-
matics is a good thing, then you cannot believe in genius anymore. The
simple consequences for the everyday work of a mathematician and, more
in general, for scienti c collaboration are left to the reader.
For these reasons, I also thank God for all these gifts and for his patience.Abstract and structure
The main aim of the present work is to start a new theory of actual in-
nitesimals, called theory of Fermat reals. After the work of A. Robinson on
nonstandard analysis (NSA), several theories of in nitesimals have been de-
veloped: synthetic di erential geometry, surreal numbers, Levi-Civita eld,
Weil functors, to cite only some of the most studied. We will discuss in
details of these theories and their characteristics, rst of all comparing them
with our Fermat reals. One of the most important di erences is the philo-
sophical thread that guided us during all the development of the present
work: we tried to construct a theory with a strong intuitive interpretation
and with non trivial applications to the in nite-dimensional di erential ge-
ometry of spaces of mappings. This driving thread tried to develop a good
dialectic between formal properties, proved in the theory, and their infor-
mal interpretations. The dialectic has to be, as far as possible, in both
directions: theorems proved in the theory should have a clear and useful
intuitive interpretation and, on the other hand, the intuition corresponding
to the theory has to be able to suggest true sentences, i.e. conjectures or
sketch of proofs that can then be converted into rigorous proofs. Almost
all the present theories of actual in nitesimals are either based on formal
approaches, or are not useful in di erential geometry. As a meaningful ex-
ample, we can say that the Fermat reals can be represented geometrically
(i.e. they can be drawn) respecting the total order relation.
The theory of Fermat reals takes a strong inspiration from synthetic
di erential geometry (SDG), a theory of in nitesimals grounded in Topos
theory and incompatible with classical logic. SDG, also called smooth in-
nitesimal analysis, originates from the ideas of Lawvere [1979] and has been
greatly developed by several categorists. The result is a powerful theory able
to develop both nite and in nite dimensional di erential geometry with a
formalism that takes great advantage of the use of in nitesimals. This the-
ory is however incompatible with classical logic and one is forced to work
in intuitionistic logic and to construct models of SDG using very elaborated
topoi. The theory of Fermat reals is sometimes formally very similar to SDG
and indeed, several proofs are simply a reformulation in our theory of the
corresponding proofs in SDG. However, our theory of Fermat reals is fully
compatible with classical logic. We can thus describe our work as a way to
bypass an impossibility theorem of SDG, i.e. a way considered as impossible
by several researchers. The di erences between the two theories are due to
our constraint to have always a good intuitive interpretation, whereas SDG
develops a more formal approach to in nitesimals.
Generally speaking, we have constructed a theory of in nitesimals which
does not need a background of logic to be understood. On the contrary,
nonstandard analysis and SDG need this non trivial background, and this is
a great barrier for potential users like physicists or engineers or even several