Fermat RealsNilpotent Infinitesimals and Infinite Dimensional Spaces
Fermat Reals
Nilpotent Infinitesimals and Infinite Dimensional Spaces
dc.contributor.advisor  Albeverio, Sergio  
dc.contributor.author  Giordano, Paolo  
dc.date.accessioned  20200415T11:11:14Z  
dc.date.available  20200415T11:11:14Z  
dc.date.issued  20.01.2010  
dc.identifier.uri  http://hdl.handle.net/20.500.11811/4509  
dc.description.abstract  The main aim of the present work is to start a new theory of actual infinitesimals, called theory of Fermat reals. After the work of A. Robinson on nonstandard analysis (NSA), several theories of infinitesimals have been developed: synthetic differential geometry, surreal numbers, LeviCivita field, Weil functors, to cite only some of the most studied. We will discuss in details of these theories and their characteristics, first of all comparing them with our Fermat reals. One of the most important differences is the philosophical 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 infinitedimensional differential geometry of spaces of mappings. This driving thread tried to develop a good dialectic between formal properties, proved in the theory, and their informal interpretations. Almost all the present theories of actual infinitesimals are either based on formal approaches, or are not useful in differential geometry. As a meaningful example, 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 differential geometry (SDG), a theory of infinitesimals grounded in Topos theory and incompatible with classical logic. SDG, also called smooth infinitesimal analysis, originates from the ideas of Lawvere and has been greatly developed by several categorists. The result is a powerful theory able to develop both finite and infinite dimensional differential geometry with a formalism that takes great advantage of the use of infinitesimals. This theory 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 differences between the two theories are due to our constraint to have always a good intuitive interpretation, whereas SDG develops a more formal approach to infinitesimals. Generally speaking, we have constructed a theory of infinitesimals which does not need a background of logic to be understood.  
dc.language.iso  eng  
dc.rights  In Copyright  
dc.rights.uri  http://rightsstatements.org/vocab/InC/1.0/  
dc.subject.ddc  510 Mathematik  
dc.title  Fermat Reals  
dc.title.alternative  Nilpotent Infinitesimals and Infinite Dimensional Spaces  
dc.type  Dissertation oder Habilitation  
dc.publisher.name  Universitäts und Landesbibliothek Bonn  
dc.publisher.location  Bonn  
dc.rights.accessRights  openAccess  
dc.identifier.urn  https://nbnresolving.org/urn:nbn:de:hbz:5N20087  
ulbbn.pubtype  Erstveröffentlichung  
ulbbnediss.affiliation.name  Rheinische FriedrichWilhelmsUniversität Bonn  
ulbbnediss.affiliation.location  Bonn  
ulbbnediss.thesis.level  Dissertation  
ulbbnediss.dissID  2008  
ulbbnediss.date.accepted  20091221  
ulbbnediss.institute  MathematischNaturwissenschaftliche Fakultät : Fachgruppe Mathematik / Mathematisches Institut  
ulbbnediss.fakultaet  MathematischNaturwissenschaftliche Fakultät  
dc.contributor.coReferee  Koepke, Peter 
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