Probability Uncertainty and Quantitative Risk最新文献

{"title":"The value does not exist! A motivation for extremal analysis","authors":"J. Aubin, H. Frankowska","doi":"10.3934/puqr.2022013","DOIUrl":"https://doi.org/10.3934/puqr.2022013","url":null,"abstract":"<p style='text-indent:20px;'>Standard mathematical economics studies the production, exchange, and consumption of goods “<i>provided with units of measurement</i>,” as in physics, in order to be enumerated, quantified, added, etc. Therefore, “baskets of goods,” which should describe subsets of goods, are mathematically represented as commodity vectors of a vector space, linear combination of units of goods, evaluated by prices, which are linear numerical functions. Therefore, in this sense, mathematical economics is a branch of physics.</p><p style='text-indent:20px;'>However, economics, and many other domains of life sciences, investigate also what will be called <i>entities</i>, defining <i>elements deprived of units of measure</i>, which thus cannot be enumerated.</p><p style='text-indent:20px;'>(1) Denoting by <inline-formula><tex-math id=\"M1\">begin{document}$X$end{document}</tex-math></inline-formula> the set of entities <inline-formula><tex-math id=\"M2\">begin{document}$x in X$end{document}</tex-math></inline-formula> <i>deprived of units of measurement</i>, a “basket of goods” is actually a <i>subset</i> <inline-formula><tex-math id=\"M3\">begin{document}$K subset X$end{document}</tex-math></inline-formula> of the set entities, i.e., an element of the “<i>hyperset</i>” <inline-formula><tex-math id=\"M4\">begin{document}${cal{P}}(X)$end{document}</tex-math></inline-formula>, the family of subsets of <inline-formula><tex-math id=\"M5\">begin{document}$X$end{document}</tex-math></inline-formula>, and no longer a commodity vector of the vector space of commodities;</p><p style='text-indent:20px;'>(2) Entities can be “gathered” instead of being “added”;</p><p style='text-indent:20px;'>(3) Entities can still be evaluated by a <i>family</i> of functions <inline-formula><tex-math id=\"M6\">begin{document}$A: x in X mapsto A(x) in mathbb{R}$end{document}</tex-math></inline-formula> regarded as a “valuators,” in lieu and place of linear “prices” evaluating the units of economic goods.</p><p style='text-indent:20px;'>(4) Subsets of entities can be evaluated by an “<i>interval of values</i>” <i>between two extremal ones</i>, the minimum and the maximum, instead of the sum of values of units of goods weighted by their quantities.</p><p style='text-indent:20px;'>Life sciences dealing with intertwined relations among many combinations of entities, hypersets offer metaphors of “Lamarckian complexity” that keeps us away from binary relations, graphs of functions, and set-valued maps, to focus our attention on “<i>multinary relations</i>” between families of hypersets. Even deprived of units of measurement, these “proletarian” entities still enjoy enough properties for this pauperization to be mathematically consistent.</p><p style='text-indent:20px;'>This is the object of this <i>extremal manifesto</i>: <i>in economics and other domains of life sciences, vector spaces should yield their imperial status of “state space” to hypersets and linear prices to hypervaluato","PeriodicalId":42330,"journal":{"name":"Probability Uncertainty and Quantitative Risk","volume":"331 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80523887","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}