Unrealistic models for realistic computations: how idealisations help represent mathematical structures and found scientific computing.
We examine two very different approaches to formalising real computation, commonly referred to as "Computable Analysis" and "the BSS approach". The main models of computation underlying these approaches—bit computation (or Type-2 Effectivity) and BSS, respectively—have also been put forward as appro...
| Publicado en: | Synthese Vol. 199; no. 1/2; pp. 249 - 284 |
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| Formato: | Artículo |
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Springer Nature
Dec2021
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| Acceso en línea: | Ver este registro en EBSCOhost |
| fields | @attributes: recordID: 1 pdfLink: plink: https://search.ebscohost.com/login.aspx?direct=true&db=hlh&AN=153650783&site=ehost-live header: @attributes: shortDbName: hlh uiTerm: 153650783 longDbName: Humanities International Complete uiTag: AN controlInfo: bkinfo: jinfo: jid: 00397857 4LI jtl: Synthese issn: 00397857 maglogo: N pubinfo: dt: Dec2021 vid: 199 iid: 1/2 pid: 237 pub: Springer Nature artinfo: ui: 153650783 10.1007/s11229-020-02654-8 ppf: 249 ppct: 35 formats: fmt: – @attributes: type: T – @attributes: type: P size: 612KB tig: atl: Unrealistic models for realistic computations: how idealisations help represent mathematical structures and found scientific computing. aug: au: Papayannopoulos, Philippos affil: Sidney M. Edelstein Centre for History and Philosophy of Science Technology and Medicine, The Hebrew University of Jerusalem, Edmund J. Safra Campus, 91904, Jerusalem, Israel su: Scientific computing Computational mathematics Floating-point arithmetic Applied mathematics Physical sciences sug: subj: Scientific computing Computational mathematics Floating-point arithmetic Applied mathematics Physical sciences keyword: Bit computation BSS/Real-RAM model Computable analysis Foundations of scientific computing Idealisations Real complexity Stability Unrealistic models Well-posed, Ill-posed physical problems ab: We examine two very different approaches to formalising real computation, commonly referred to as "Computable Analysis" and "the BSS approach". The main models of computation underlying these approaches—bit computation (or Type-2 Effectivity) and BSS, respectively—have also been put forward as appropriate foundations for scientific computing. The two frameworks offer useful computability and complexity results about problems whose underlying domain is an uncountable space (such as R or C ). Since typically the problems dealt with in physical sciences, applied mathematics, economics, and engineering are also defined in uncountable domains, it is fitting that we choose between these two approaches a foundational framework for scientific computing. However, the models are incompatible as to their results. What is more, the BSS model is highly idealised and unrealistic; yet, it is the de facto implicit model in various areas of computational mathematics, with virtually no problems for the everyday practice. This paper serves three purposes. First, we attempt to delineate what the goal of developing foundations for scientific computing exactly is. We distinguish between two very different interpretations of that goal, and on the separate basis of each one, we put forward answers about the appropriateness of each framework. Second, we provide an account of the fruitfulness and wide use of BSS, despite its unrealistic assumptions. Third, according to one of our proposed interpretations of the scope of foundations, the target domain of both models is a certain mathematical structure (namely, floating-point arithmetic). In a clear sense, then, we are using idealised models to study a purely mathematical structure (actually a class of such structures). The third purpose is to point out and explain this intriguing (perhaps unique) phenomenon and attempt to make connections with the typical case of idealised models of empirical domains. pubtype: Academic Journal doctype: Article src: R language: English refInfo: copyright: @attributes: flag: Y custom: Synthese is a copyright of Springer, 2021. All Rights Reserved. item: Synthese holder: Springer Nature dt: @attributes: year: 2021 holdings: @attributes: islocal: N |
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