What is the core of Computing? This paper defines the discipline of computing as centered around the notion of modeling, especially those models that are automatable and automatically manipulable. We argue that this central idea crucially connects models with languages and machines rather than focusing on and around computational artifacts, and that it admits a very broad set of fields while still distinguishing the discipline from mathematics, engineering and science. The resulting computational curriculum focuses on modeling, scales and limits, simulation, abstraction, and automation as key components of a computationalist mindset.
During the early 1830's Bernard Bolzano, working in Prague, wrote a manuscript giving a foundational account of numbers and their properties. In the final section of his work he described what he called 'infinite number expressions' and 'measurable numbers'. This work was evidently an attempt to provide an improved proof of the sufficiency of the criterion usually known as the 'Cauchy criterion' for the convergence of an infinite sequence. Bolzano had in fact published this criterion four years earlier than Cauchy who, in his work of 1821, made no attempt at a proof. Any such proof required the construction or definition of real numbers and this, in essence, was what Bolzano achieved in his work on measurable numbers. It therefore pre-dates the well-known constructions of Dedekind, Cantor and many others by several decades. Bolzano's manuscript was partially published in 1962 and more fully published in 1976. We give an account of measurable numbers, the properties Bolzano proved about them, and the controversial reception they have prompted since their publication.
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