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**Mathematical structures and physical necessity.**
*(English)*
Zbl 0847.00006

Echeverria, Javier (ed.) et al., The space of mathematics. Philosophical, epistemological, and historical explorations. Revised papers from a symposium on structures in mathematical theories, Donostia/San Sebastian, Basque Country, Spain, September 1990. Berlin: Walter de Gruyter. Grundlagen der Kommunikation und Kognition. 132-140 (1992).

Two forms of perceived necessity in mathematical physics are distinguished: “present necessity” or necessity of configurations, and “future necessity” or necessity of processes. We will perceive the former if we try to tile a floor with regular pentagons of the same size, realizing that this is impossible. An example for the latter can be perceived by watching a match catching a piece of paper and turning it to ashes.

A configurational necessity is understood if its conditions are overseeable and if the necessity of a feature perceived is grasped “as a necessary consequence of the characteristics by which that configuration is conceived” (p. 134). The understanding of the necessity of processes results from analogies to geometric necessity: “the conception of time – past, present, and future – as a single, homogeneous linear continuum” (p. 135). The author discusses ancient Greek mappings of time into space, compares them with procedures in Newtonian physics, and maintains that such mathematical representations of natural processes can even answer Kant’s philosophical question “How shall I understand that because something exists something else should exist as well” (pp. 137-138).

The author ends with considerations on reasons for the missing certainty for forecasts, the relation between forecast and understanding, and the “perhaps most significant feature of the mathematical representation of nature, […] that the exact concepts of mathematics must be blurred in order to fit our observations” (p. 139), referring to G. Ludwing’s theory of such blurrings [Deutung des Begriffs “physikalische Theorie” und axiomatische Grundlegung der Hilbertraumstruktur der Quantenmechanik durch HauptsĂ¤tze des Messens (Lecture Notes on Physics, 4; Springer, Heidelberg) (1970; Zbl 0204.29402)].

For the entire collection see [Zbl 0839.00019].

A configurational necessity is understood if its conditions are overseeable and if the necessity of a feature perceived is grasped “as a necessary consequence of the characteristics by which that configuration is conceived” (p. 134). The understanding of the necessity of processes results from analogies to geometric necessity: “the conception of time – past, present, and future – as a single, homogeneous linear continuum” (p. 135). The author discusses ancient Greek mappings of time into space, compares them with procedures in Newtonian physics, and maintains that such mathematical representations of natural processes can even answer Kant’s philosophical question “How shall I understand that because something exists something else should exist as well” (pp. 137-138).

The author ends with considerations on reasons for the missing certainty for forecasts, the relation between forecast and understanding, and the “perhaps most significant feature of the mathematical representation of nature, […] that the exact concepts of mathematics must be blurred in order to fit our observations” (p. 139), referring to G. Ludwing’s theory of such blurrings [Deutung des Begriffs “physikalische Theorie” und axiomatische Grundlegung der Hilbertraumstruktur der Quantenmechanik durch HauptsĂ¤tze des Messens (Lecture Notes on Physics, 4; Springer, Heidelberg) (1970; Zbl 0204.29402)].

For the entire collection see [Zbl 0839.00019].

Reviewer: V.Peckhaus (Erlangen)