6 research outputs found
Podstawy matematyki bez aktualnej nieskończoności
Contemporary mathematics significantly uses notions which belong to ideal
mathematics (in Hilbert’s sense) – which is expressed in language which essentially
uses actual infinity. However, we do not have a meaningful notion
of truth for such languages. We can only reduce the notion of truth to finitistic
mathematics via axiomatic theories. Nevertheless, justification of truth
of axioms themselves exceeds the capabilities of the theory based on these
axioms.
On the other hand, we can easily decide the truth or falsity of a statement
in finite structures. The aim of this dissertation is to identify the fragment
of mathematics, which is of the finitistic character. The fragment of mathematics
which can be described without actual infinity. This is the part of
mathematics which can be described in finite models and for which the truth
of its statements can be verified within finite models.We call this fragment of
mathematics with a term introduced by Knuth – the concrete mathematics.
This part of mathematics is of computational character and it is closer to
our empirical base, which makes it more difficult.
We consider concrete foundations of mathematics, in particular the concrete
model theory and semantics without actual infinity. We base on the
notion of FM–representability, introduced by Mostowski, as an explication of
expressibility without actual infinity. By the Mostowski’s FM–representability
theorem, FM–representable notions are exactly those, which are recursive
with the halting problem as an oracle.
We show how to express basic concepts of model theory in the language
without actual infinity. We investigate feasibility of the classical model–
theoretic constructions in the concrete model theory. We present the Concrete
Completeness Theorem and the Low Completeness Theorem; the Concrete
Omitting Types Theorem; and Preservation Theorems. We identify
the constructions which are not admissible in the concrete model theory by
showing stages of these constructions which are not allowed in the concrete
framework. We show which arguments from the axiomatic model theory fail
in the concrete model theory.
Moreover, we investigate how to approximate truth for finite models.
In particular we study the properties of approximate FM–truth definitions
which are expressible in modal logic. We introduce modal logic SL, axioms
of which mimic the properties of a specific approximate FM–truth definition.
We show that SL is the modal logic of any approximate FM–truth definition.
This is done by proving a theorem analogous to Solovay’s completeness
theorem for modal logic GL.Współczesna matematyka w znaczącej mierze posługuje się pojęciami, które należą do matematyki idealnej (w sensie Hilberta) -- wyrażona jest w języku istotnie wykorzystującym aktualną nieskończoność. Dla tego typu języków nie posiadamy sensownego kryterium prawdziwości. Jesteśmy w stanie jedynie redukować je do matematyki skończonościowej poprzez teorie aksjomatyczne. Niemniej uzasadnianie prawdziwości samych aksjomatów znajduje się poza zasięgiem teorii na nich opartej.
Z drugiej strony w strukturach skończonych jesteśmy w stanie w prosty sposób rozstrzygać prawdziwość i fałszywość twierdzeń. Celem niniejszej rozprawy jest identyfikacja fragmentu matematyki, który ma skończonościowy charakter. Fragmentu matematyki, do którego opisu nie jest niezbędna aktualna nieskończoność, a wystarczy jedynie nieskończoność potencjalna. Jest to ta część matematyki, której pojęcia można wyrazić w modelach skończonych oraz prawdziwość twierdzeń której można w nich zweryfikować. Tę część matematyki, za Knuthem, nazywamy matematyką konkretną. Ma ona obliczeniowy, kombinatoryczny charakter i jest bliższa naszemu doświadczeniu niż matematyka idealna, a co za tym idzie jest trudniejsza.
Rozważamy konkretne podstawy matematyki, w szczególności konkretną teorię modeli oraz semantykę bez aktualnej nieskończoności. Opieramy się na wprowadzonym przez Mostowskiego pojęciu FM--reprezentowalności, jako eksplikacji wyrażalności bez aktualnej nieskończoności oraz twierdzeniu o FM--reprezentowalności identyfikującym FM--reprezentowalne pojęcia z tymi, które są obliczalne z problemem stopu jako wyrocznią.
Pokazujemy w jaki sposób można zinterpretować podstawowe pojęcia teorii modeli w języku bez aktualnej nieskończoności. Następnie badamy klasyczne konstrukcje teoriomodelowe pod kątem ich wykonalności w obszarze matematyki konkretnej. Prezentujemy twierdzenie o konkretnej pełności oraz twierdzenie o łatwej pełności, twierdzenie o omijaniu typów oraz twierdzenia o zachowaniu. Przedstawiamy konstrukcje, które są niewykonalne dla modeli konkretnych, identyfikując etapy konstrukcji teoriomodelowych, które nie są wykonalne w teorii modeli konkretnych. Identyfikujemy argumenty z aksjomatycznej teorii mnogości, które nie są dopuszczalne w konkretnej teorii modeli.
Ponadto, badamy możliwość przybliżania prawdy arytmetycznej w modelach skończonych. W szczególności rozważamy te własności przybliżonych predykatów prawdy dla modeli skończonych, które wyrażalne są w logice modalnej. Wprowadzamy logikę modalną SL, której aksjomaty odzwierciedlają własności przybliżonych predykatów prawdy. Pokazujemy, że logika SL jest logiką przybliżonych predykatów prawdy -- dowodzimy twierdzenia analogicznego do twierdzenia o pełności dla logiki GL udowodnionego przez Solovaya
Searching the space of representations: reasoning through transformations for mathematical problem solving
The role of representation in reasoning has been long and widely regarded as crucial.
It has remained one of the fundamental considerations in the design of information-processing
systems and, in particular, for computer systems that reason. However, the
process of change and choice of representation has struggled to achieve a status as a
task for the systems themselves. Instead, it has mostly remained a responsibility for
the human designers and programmers.
Many mathematical problems have the characteristic of being easy to solve only
after a unique choice of representation has been made. In this thesis we examine two
classes of problems in discrete mathematics which follow this pattern, in the light of
automated and interactive mechanical theorem provers. We present a general notion of
structural transformation, which accounts for the changes of representation seen in such
problems, and link this notion to the existing Transfer mechanism in the interactive
theorem prover Isabelle/HOL.
We present our mechanisation in Isabelle/HOL of some specific transformations identified as key in the solutions of the aforementioned mathematical problems. Furthermore,
we present some tools that we developed to extend the functionalities of the
Transfer mechanism, designed with the specific purpose of searching efficiently the
space of representations using our set of transformations. We describe some experiments
that we carried out using these tools, and analyse these results in terms of how
close the tools lead us to a solution, and how desirable these solutions are.
The thorough qualitative analysis we present in this thesis reveals some promise as
well as some challenges for the far-reaching problem of representation in reasoning, and
the automation of the processes of change and choice of representation
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Machine-Learning and Data Science Techniques in String and Gauge Theories
Techniques from supervised and unsupervised machine learning, along with those from data and network science, are applied to generated datasets of mathematical objects relevant to string and gauge theories. Investigations show success in identifying and learning new structure associated to these objects. Datasets considered in the research work completed for this thesis include: dessins d’enfants, quivers, Hilbert series, amoebae, polytopes, Calabi-Yau manifolds, brane webs, and cluster algebras
Iannis Xenakis and Sieve Theory: An Analysis of the Late Music (1984-1993)
This thesis is divided in three parts, the first two of which are theoretical and the third analytical. Part I is an investigation of lannis Xenakis's general theory of composition, the theory of outside-time musical structures. This theory appears in many of Xenakis's writings, sometimes quite idiosyncratically. The aim of this part is to reveal the function of the non-temporal in Xenakis's musical structures, by means of a historical approach through his writings. This exploration serves to unveil certain aspects discussed more thoroughly through a deconstructive approach. The deconstructive is demonstrated in the classification of musical structures and aims partly at showing the nature of Time in Xenakis's theory.
Part II is preoccupied with Xenakis's Sieve Theory. In the earlier writings on Sieve Theory he presented a slightly different approach than in the later, where he also provided an analytical algorithm that he developed gradually from the mid1980s until 1990. The rationale of this algorithm and the pitch-sieves of 1980-1993 guides Part III, which is preoccupied with a methodology of sieve analysis, its application, and an exploration of the employment of sieves in some of Xenakis's compositions of the 1980s. When possible, the analysis takes in consideration the pre-compositional sketches, available at the Archives Xenakis, Bibliotheque Nationale de France. The sketches reveal aspects of the application of Sieve Theory, not included in Xenakis's theoretical writings.
As with the application of other theories, Xenakis progressed to less formalised processes. However, this does not mean that Sieve Theory ceased to inform the process of scale-construction. As the conclusion of this dissertation indicates, he employed Sieve Theory in order to achieve structures that conform to his general aesthetic principles that relate to various degrees of symmetry and periodicity