Why More Is Different by Brigitte Falkenburg & Margaret Morrison

Author:Brigitte Falkenburg & Margaret Morrison
Language: eng
Format: epub
Publisher: Springer Berlin Heidelberg, Berlin, Heidelberg

8.5 The Axiomatic Method and Reduction

An ideal of long standing in the physical sciences is to represent the fundamental principles of a scientific field in axiomatic form and to take those principles as implicitly containing all of the knowledge in that field. If the first principles constitute a complete theory—and what ‘complete’ means in the context of scientific theories is a tangled issue—then the only role for empirical data is to provide the initial or boundary conditions for the system. Of course, testing and confirming the theory will require contact with data reports as well, but if the axioms are true, that aspect can be set aside. This ideal situation is unavailable in many cases and three consequences of its absence are of especial concern for us.

The first occurs when the values of certain parameters must be known in order for predictive use to be made of the theory. Borrowing a term from quantum chemistry, call a deductive or computational method semi -empirical if, in addition to the fundamental principles and parameters for the domain in question, some non-fundamental facts about the system must be estimated from empirical data rather than calculated from the fundamental principles and parameters. A simple example is when the value of the elasticity parameter in Hooke’s Law cannot be estimated from first principles and must be known on the basis of measurement. Semi-empirical methods are often forced on us because of practical limitations but the results discussed in this paper show that, if these models, or ones like them, accurately represent systems in the physical world, then semi-empirical methods are unavoidable in areas beyond their origins in chemistry and this puts essential limits on constructivist knowledge..

The second consequence is that this situation places limits on the scope of theoretical knowledge and reveals an additional role for empirical input into scientific representations.17 This consequence of undecidability has long been discussed in the philosophy of mathematics but it must be dealt with differently in science. The data needed to supplement the fundamental theory do not originate in intuition, as their correlates in mathematics are often supposed to do, but in measurement and other empirical procedures. This places them on a much less controversial epistemological basis. In order to know these macroscopic values we must either measure them directly or calculate them using a theory that represents the fundamental facts within a different conceptual framework.

The third consequence is that these results show limits on the hypothetico-deductive method. Suppose that we have a scientific theory T that is true of some states of a system but not of others and that all of the sentences that would falsify T fall into the class of sentences that are undecidable with respect to T. When we obtain empirical data and find that it is consistent with, for example, the truth conditions for a sentence ¬S but not with S, but neither S nor ¬S can be derived from T, that data cannot falsify T. S and ¬S are still falsifiable sentences

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