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2014 | OriginalPaper | Chapter

5. Evolutions and Their Temporal Windows

Author : Jean-Pierre Aubin

Published in: Time and Money

Publisher: Springer International Publishing

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Abstract

We did not respect in the illustrations presented in Chaps. 2, p. 31, and 3, p. 47, the venerable tradition inherited from physical sciences studying mainly evolutions on given temporal windows [0, T] or [0, + ∞[ governed by differential systems starting at the origin 0 of the chronological time at specific initial conditions up to the future end.

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previous chapter Uncertain Endowments and Economic Cycles
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Footnotes
1
Adjective coined in 1882 by Robert Kelly, the anonymous author of [93, The Alternative: a Study in Psychology], which survived only because William James popularized it in [91, Principles of Psychology] (see [2, Andersen & Grush]). It has and will run much ink with the progress of cognitive science.
 
2
“What was God doing before He created the Heavens and the Earth?” asked Augustine of Hippo in his confessions. Is his eternity only forward in time and not backward? Introducing the concepts of temporal windows and duration function bypasses the question of origin of time.
 
3
As Dobbs proposed in 1951, instead of the usual representation by a point of the line. This philosopher was motivated by the mathematics of his time, and chose metaphors based on quantum physics and the uncertainty principle ([58, 59, Dobbs], [79, Grünbaum], [39, Broad]) to give some “thickness” to “transitory time” by introducing a second component of the time, the “extensional time” to give a formal sense the specious present. To our knowledge, these attempts were not pursued. I thank Jean-Philippe Terreaux for having mentioned this reference.
 
4
The age-structured standard approach starts with the establishment of the McKendrick partial differential equation relating the population and its partial derivatives with respect to time and age. They do not involve explicitly calendar evolutions, except in some underlying proofs. Age-structured partial differential equations involving both time and age have been studied (see, among an extensive literature, [3, Anita], [14, Aubin], [16, Aubin, Bonneuil & Maurin], [90, Iannelli],[94, Applied Mathematical Demography], [95, Keyfitz N. & Keyfitz B.], [122, Von Foerster], [124, Webb], etc.).
 
5
That may be called durances: Efim Galperin taught me that duration is, in English, constant, so that variable duration is an oxymoron. He also told me that there exists a synonym in English, durance, which could be used for duration functions. It is derived from Old French (see the definition: DURABLETË; durance s. f. durée, perpétuité. Prov. durabletât. It. durabilità of [36, Glossaire de la langue d’oïl,(xiième-xiv ième siècles)] by Alphonse Bos and durableté, -ance, n. f. (xiiième-xv ième): 1. Durée: “ la nature divine laquelle … donne durance, gloire, honneur ” (ième siècle, Jean Le Bel); 2. Endurance. Étymologie Durable, Durer of [77, Dictionnaire d’ancien français. MoyenÂge et Renaissance] by Robert Grandsaignes d’Hauterive), that he suggested to use in English and to resuscitate in French.
 
6
Some physicists would suggest that “entropy” could be a candidate to the role of duration function. This is not the place to open this voluminous Pandora Box.
 
7
The instantaneous duration function d  + ∞ ( ⋅) is no longer a single-valued map, but either the set-valued map equal to 0 for t < T and [0, D] for t = T, the graphical derive of which at (T, 0) is infinite or else, the Heavyside distribution, the derivative of which is the Dirac measure at T. It is not a member of \(\mathcal{A}_{{[\varphi }^{\flat }{,\varphi }^{\sharp }]}(T,D)\) and we shall not use them in this study.
 
8
Among so many Maupertuis contributions, see [106‐109, Maupertuis].
 
9
Without entering philosophical considerations, following the paradoxes of Zeno of Elea, we may observe, as Bergson pointed out in [32, 33, Creative Evolution], that the perception of a continuous evolution of time is fallacious, that time reduces to a “dust of instants” (poussière d’instants). Measuring chronological time would require to stop it for taking a (static) measure. But measures do not need to be static: assuming that the time is a numéraire of evolution, it can be dynamically measured by clocks “reproducing” evolutions with velocity equal to 1.
 
10
Clio, muse of history, was born as the other muses out of the love between Zeus and Mnenosyne, Goddess of memory.
 
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Metadata
Title
Evolutions and Their Temporal Windows
Author
Jean-Pierre Aubin
Copyright Year
2014
Book
Time and Money

Print ISBN: 978-3-319-00004-6

Electronic ISBN: 978-3-319-00005-3

Copyright Year: 2014

DOI
https://doi.org/10.1007/978-3-319-00005-3_5