The value does not exist! A motivation for extremal analysis
Authors: Aubin Jean-Pierre , Frankowska Hélène 
Published: October 2022

<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 hypervaluators</i>.</p><p style='text-indent:20px;'>We no longer have to add goods which can only be gathered, prices do not have to be linear, although it costs some effort to deprive oneself of the powerful and luxurious charms of convex and linear functional analysis motivated by physics.</p><p style='text-indent:20px;'>These sacrifices concern only economics and other fields of life science, since physicists deal with experimental observations of objects endowed with unit of measurement by adequate processes of measurement. They can happily live in vector spaces without any guilt. This is not the case of life scientists, who have mainly history to support and validate their observations, with, sometimes, the privilege to statistically measure the frequency of some of them.</p>