Mathematics as a Tool: Tracing New Roles of Mathematics in the Sciences by Johannes LenhardMy rating: 4 of 5 stars
...Expounding on the philosophy of model building with several historical examples, this collection of papers highlights approaches to modeling and underpinning necessities. Among the necessities highlighted is an overview of theory of error analysis, a few diagrams of the necessary high-level modeling steps, and transforming the model into mathematical terms (“idealizations play a central role in this transformation”). Modeling is an obvious application of the mathematics tool. This common instance of mathematics employed in science falls into the more general umbrella summarized by contributor Jürgen Jost that mathematics “consists in providing a framework for conceptual thinking and formal reasoning. This is valid across all disciplines, and actually badly needed in many of them.” Jost’s concluding contribution nicely points to what the considerations contained herein support: “there exist profound analogies between models in physically very different domains. It is actually one of the main driving forces of mathematics to consider the corresponding structures and relations abstractly and independently of any particular instantiation and to work out general theories to new domains… For instance, similar statistical phenomena show up in quantum mechanics, in the analysis of biological high-throughput data, or in the description of social phenomena.” Mathematics, one tool for all science.
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