<commit_before>% !TEX root = thesis.tex

\chapter{Mathematical foundations of spatial modelling}
\label{ch:modelling-mathematics}

Spatial modelling has its origins in the geographical notion of space, which is in turn based on our own observations of the world and empirical experience \citep{Couclelis99}.
However, these informal notions are error-prone and differ from person to person.
In order to describe space unambiguously, people have thus turned to models that still describe geographical phenomena, but do so using formal notions derived from mathematics.
These formal models make it possible to create and store digital representations of the world in a computer, and thus to use the power of a computer to easily solve spatial problems \citep{Burrough86,Bailey95}.

The current chapter describes some of these formal notions and their relevant context, which are used to study the spatial modelling approaches presented in the upcoming chapters.
% \refse{se:settheory} introduces the main concepts of set theory and mathematical logic, the branches of mathematics that are thereafter used as a basis for more complex notions.
\refse{se:geometry} introduces geometry, which provides the basic concepts used to describe the position, shape and orientation of objects.
\refse{se:topology} builds on these to present topology, which formalises notions such as the boundary and interior of an object or the relationships between multiple objects.

% \section{Set theory and mathematical logic}
% \label{se:settheory}

% Set theory is the branch of mathematics that studies \emph{sets}, which are collections of abstract objects.
% While the study of set theory only formally started with \citet{Cantor74}, its intuitive and minimal concepts were later used in order to give a foundation to almost all areas of mathematics\footnote{Even as some mathematicians and philosophers have argued against set theory as a foundation for all of mathematics.}.

% Set theory starts by considering the existence of a given domain of objects from which one may build sets, which is known as the \emph{universe set} and denoted as $\mathbb{U}$.
% These objects can be anything, including other sets.<commit_msg>Small fix based on Roderik's comments<commit_after>% !TEX root = thesis.tex

\chapter{Mathematical foundations of spatial modelling}
\label{ch:modelling-mathematics}

Spatial modelling has its origins in the geographical notion of space, which is in turn based on our own observations of the world and empirical experience \citep{Couclelis99}.
However, these informal notions are error-prone and differ from person to person.
In order to describe space unambiguously, people have thus turned to models that still describe geographical phenomena, but do so using formal notions derived from mathematics.
These formal models make it possible to create and store digital representations of the world in a computer, and thus to use the power of a computer to easily solve spatial problems \citep{Burrough86,Bailey95}.

The current chapter describes some of these formal notions and their relevant context, which are used to study the spatial modelling approaches presented in the upcoming chapters.
% \refse{se:settheory} introduces some concepts of elementary set theory and mathematical logic, which are later used in definitions in this thesis.
\refse{se:geometry} introduces geometry, which provides the basic concepts used to describe the position, shape and orientation of objects.
\refse{se:topology} builds on these to present topology, which formalises notions such as the boundary and interior of an object or the relationships between multiple objects.

% \section{Set theory and mathematical logic}
% \label{se:settheory}

% Set theory is the branch of mathematics that studies \emph{sets}, which are collections of abstract objects.
% While the study of set theory only formally started with \citet{Cantor74}, its intuitive and minimal concepts were later used in order to give a foundation to almost all areas of mathematics\footnote{Even as some mathematicians and philosophers have argued against set theory as a foundation for all of mathematics.}.

% Set theory starts by considering the existence of a given domain of objects from which one may build sets, which is known as the \emph{universe set} and denoted as $\mathbb{U}$.
% These objects can be anything, including other sets.
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