|Zusammenfassung||This dissertation consists of two parts, related in that the first part motivates the mathematical questions addressed by the second.
The first part treats the concepts vulnerability (to climate change) and probability, as well as the method of formalization. Formalization is here considered to be a translation of concepts from everyday language or scientific terminologies into mathematical concepts. The result of such a formalization is a mathematical model of a concept. An advantage of using mathematical concepts is that they do not have implicit connotations.
The concept probability, with its different interpretations and mathematical models, is considered as an example of formalization. A subjective probability interpretation is of interest for climate change related research because, unlike the frequentist interpretation, it makes probability applicable also in cases where long data series are not available. The subjective interpretation is presented using de Finetti's model of coherent probability assignments. Different mathematical models of probability are related to the concept itself in differing ways: de Finetti's model explains its meaning and provides a rule for measuring probabilities, whereas Kolmogorov's model mainly serves as a basis for the mathematical theory; it specifies neither meaning nor measurement of probability. All in all, the example concept probability shows that formalization is not a miracle cure against conceptual unclarity, but that it can support clarification.
Conceptual clarification was one aim of a formalization of vulnerability and related concepts developed at the Potsdam Institute for Climate Impact Research. This work presents the formalization and uses it for clarifying a terminology for which the literature states a 'Babylonian confusion'. Many similar theoretical definitions of vulnerability exist, but there are different interpretations and approaches to assessing it.
With the help of precise but nevertheless general mathematical definitions of vulnerability, the common structure of many definitions from the literature is verified and two assessment approaches are distinguished. Several pairs of vulnerability interpretations from the literature, as well as types of risk assessments in the context of natural hazards, are traced back to this distinction of assessment approaches. The conceptual confusion is explained by the finding that there is no one-to-one mapping from technical terms to assessment approaches, nor vice versa.
Mathematics as a `lingua franca' is of limited scope in the field of vulnerability research, which is dominated by the social sciences. To provide the formalization in a more accessible format for non-mathematicians, it is also represented by diagrams.
The first part of this work considers mathematics as a language, discusses formalization as translation into mathematics (using the concept probability), and translates vulnerability into this language, thus producing clarity. Finally, it discusses how mathematics can be presented in a generally agreeable manner. It can thus be considered a work about mathematics.
The second part, on the other hand, is a work in mathematics in a stricter sense: it combines elements from category theory and probability theory. The mathematical model of vulnerability presented by Ionescu  uses the category theoretical concepts of a functor and a monad to generically describe the uncertain future evolution of a system. Probability is one way of describing uncertainty mathematically; for probability measures in the sense of Kolmogorov's axioms a functor and monad exist, as was shown by Giry . This work establishes that also finitely additive probabilities and coherent probability assignments form functors on the category of sets and functions. Further, several monads of finitely additive probabilities are identified. This is a first step towards using these more general models of (subjective) probability in the field of vulnerability research.