Geologische Rundschau

, Volume 77, Issue 2, pp 591–607 | Cite as

Geology and mathematics: Progressing mathematization of geology

  • H. Schaeben
Article

Abstract

Surveying the history of mathematical geology since the times of Lyell it is shown that its characteristic feature is that of interaction between a strongly historically inclined science and rather abstract mathematics/statistics. It is proven that mathematization of geology and experimental geology stimulate one another, and that mathematical geology can be of essential aid in formulating conceptual models and scientific theories to integrate and unify diverse geological phenomena.

Examples of the progressing mathematization of geology and the geosciences have been chosen to be most instructive for the purposes of understanding the general law of the development of science in the geosciences, as well as the propagation of mathematical models and their numerical realization. They range from almost conventional application of classical statistics in subdividing the Tertiary, to mathematical analysis of directional and orientation data vital to plate tectonics, deterministic and stochastic approaches for modeling and simulation purposes in characterization and management of natural resources, and the application of bifurcation theory to study differentiated layering as well as research in artificial intelligence and expert systems in exploration. All examples will be briefly presented and discussed, in general terms, avoiding all severe mathematics. Similarities and differences in the lawful development of geology, biology, and physics with respect to their mathematization are mentioned.

Keywords

Plate Tectonic Stochastic Approach Lawful Development Bifurcation Theory Numerical Realization 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Zusammenfassung

Die Geschichte der mathematischen Geologie seit Lyell überschauend wird festgestellt, da\ ihr charakteristisches Merkmal das der Interaktion einer historisch orientierten Naturwissenschaft mit auf Abstraktion zielender Mathematik/Statistik ist. Es wird belegt, da\ Mathematisierung der Geologie und experimentelle Geologie sich gegenseitig vorantreiben und da\ mathematische Geologie zur Bildung von konzeptionellen Modellen und wissenschaftlichen Theorien, die unterschiedliche geologische Erscheinungen in Zusammenhang stellen, wesentlich beitragen kann.

Die hier vorgestellten Beispiele der fortschreitenden Mathematisierung der Geologie und der Geowissenschaften als Ganzes sind nach dem Kriterium ausgewählt, die allgemeine Gesetzmä\igkeit des Entwicklungsprozesses der Wissenschaften für die geologischen Wissenschaften besonders deutlich darzustellen, und mit der Absicht, mathematische Modelle und ihre numerische Realisierung zu verbreiten. Sie reichen von fast standardmä\iger Anwendung klassischer, statistischer Argumente zur Unterteilung des Tertiärs über die mathematische Analyse von Richtungs- und Orientierungsdaten, welche wesentlich zur Akzeptanz der Plattentektonik beitrug, und deterministische und stochastische Zugänge zur Modellierung und Simulation bei der Charakterisierung und Verwaltung natürlicher Ressourcen bis zur Anwendung der mathematischen Verzweigungstheorie bei der Untersuchung von differenziertem metamorphem Lagenbau und zur Forschung auf dem Gebiet der künstlichen Intelligenz und der Expertensysteme in der Exploration. Die Beispiele werden ohne mathematische Formulierungen diskutiert. Auf ähnlichkeiten und Unterschiede bei der gesetzmä\igen Entwicklung von Geologie, Biologie und Physik in bezug auf ihre Mathematisierung wird hingewiesen.

Résumé

Si on résume l'histoire de la géologie mathématique depuis Lyell, on constate que son trait caractéristique réside dans l'interaction d'une science (naturelle) à orientation historique avec une mathématique-statistique tendant à l'abstraction. Il est de fait que la mathématisation de la géologie et la géologie expérimentale se stimulent mutuellement et que la géologie mathématique peut aider de manière significative à l'élaboration de modèles conceptuels et de théories scientifiques qui tendent à intégrer et à unifier les divers phénomènes géologiques.

Les exemples présentés ici de ce processus de mathématisation progressive de la géologie et des sciences de la Terre ont été choisis pour Être les plus significatifs possible, afin d'en éclairer le développement et de justifier en mÊme temps le dessein de propager l'usage des modèles mathématiques et leur réalisation numérique. Le premier exemple concerne l'application, presque conventionnelle, de la statistique classique à la subdivision du Tertiaire. On poursuit par l'analyse mathématique des données de direction et d'orientation, vitales dans l'étude du modèle de la tectonique des plaques. Viennent ensuite les approches déterministes et stochastiques de l'élaboration et de la simulation en vue de caractériser et de gérer les ressources naturelles. Les exemples se poursuivent par l'application de la théorie mathématique de la bifurcation à l'étude du rubanement métamorphique et à la recherche dans le domaine de l'intelligence artificielle et des systèmes d'experts appliqués à l'exploration. Tous les exemples sont brièvement présentés et discutés, en termes généraux, à l'exclusion de formulation mathématique. On souligne les ressemblances et les différences entre les lois du développement de la géologie, de la biologie et de la physique en regard de leur mathématisation.

кРАткОЕ сОДЕРжАНИЕ

ИстОРИь МАтЕМАтИЧЕс кОИ гЕОлОгИИ сО ВРЕМЕ НИ Lyell'a хАРАктЕРИжУЕтсь Вж АИМОДЕИстВИЕМ ЕстЕс тВОжНАНИь И ИстОРИЧЕскИ АБстРА ктНО ОРИЕНтИРОВАННы х НАУк, кАк МАтЕМАтИкА И стАт ИстИкА.

НА пРИМЕРАх ДЕМОстРИ РУЕтсь жНАЧЕНИЕ МАтЕ МАтИкИ Дль ЁкспЕРИМЕНтАльН ОИ гЕОлОгИИ И пРОНИкН ОВЕНИЕ ЁтИх МАтЕМАтИЧЕскИх НАУк В гЕОлОгИУ, ЧтО сп ОсОБстВУЕт сОжДАНИУ МОДЕлЕИ И тЕ ОРИИ пРИ ИжУЧЕНИИ РАж лИЧНых гЕОлОгИЧЕскИх ьВлЕН ИИ И т.О. РАжРЕшАЕт ИжУЧ Ить цЕлыИ РьД ЁтИх гЕОлОг ИЧЕскИх ьВлЕНИИ. УспЕ шНОЕ пРИМЕНЕНИЕ ЁтИх МАтЕ МАтИЧЕскИх МЕтОДОВ Д ль ИжУЧЕНИь цЕлОгО РьДА гЕОлОгИЧ ЕскИх жАкОНОМЕРНОст ЕИ В гЕОлОгИИ пОДЧЕРкИВАЕтсь. пРИВ ЕДЕН цЕлыИ РьД МОДЕлЕ И, кОтОРыЕ спОсОБстВУУ т пРИМЕНЕНИУ Их В РАжл ИЧНых ОБлАстьх гЕОлОгИИ. НА пР.: ИжУЧЕНИЕ тЕктОНИк И плИт с пОМОЩьУ РАжРАБОтАНН ых МОДЕлЕИ, А тАкжЕ пРИ МЕНЕНИЕ ЁтИх МОДЕлЕИ пРИ Ёксп лОРАцИИ НА РАжлИЧНыЕ пОлЕжНыЕ ИскОпАЕМыЕ. ОпИсАННы Е жДЕсь пРИМЕРы ДАУтсь БЕж пРИМЕНЕНИ ь слОжНОгО МАтЕМАтИЧ ЕскОгО АппАРАтА.

В кОНцЕ пРИВЕДЕНы пРИ МЕРы пРИМЕНЕНИь МАтЕ МАтИкИ В ОБлАстИ гЕОлОгИИ, БИ ОлОгИИ И ДР. ЕстЕстВЕН Ных НАУк.

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Copyright information

© Ferdinand Enke Verlag Stuttgart 1988

Authors and Affiliations

  • H. Schaeben
    • 1
  1. 1.Department of GeologyBonn UniversityBonn 1West Germany

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