Abstract
It might be folly to insist on one overall official characterization of the way scientific and mathematical theories ought to be presented. They are usually presented of course in various ways so that the authors and scholars in a field can find optimal means for communication with others, whether within or outside that field. Nevertheless, if theories were thought of as representations of our knowledge, then one way of looking at the representation of theories is to ask, as Hermann Weyl did (2009) “What is the ultimate purpose of forming theories?”, and he then cited the familiar proposal of Heinrich Hertz (1894):This powerful and compact statement of Hertz, resonated with David Hilbert’s revolutionary proposal for the axiomatic representation of theories. The kind of axiomatization Hilbert advocated was exemplified in his Foundations of Geometry, [1899], and was to become the model for his inquiries into the physical sciences as well as mathematical ones.
These considerations induce us to conceive of an axiom system as a logical mold (‘Leerform”) of possible sciences. A concrete interpretation is given when designata have been exhibited for the names of the basic concepts, on the basis of which the axioms become true propositions.
Hermann Weyl (Philosophy of Mathematics and Natural Science, Princeton University Press, 1949, p. 25).
Thus the axiom system itself does not express something factual; rather, it presents only a possible form of a system of connections that must be investigated mathematically according to its internal [innere] properties.
Paul Bernays (“Hilbert’s Significance for the Philosophy of Mathematics”(1922) tr. By P. Mancosu, From Brouwer to Hilbert, Oxford University Press, 1998, p. 192).
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Notes
- 1.
We should note that some translations other than “scaffolding” are a great improvement, though the strict meaning of “Fachverk” is not used. W. Ewald, in his translation (1996) of Hilbert’s Axiomatic Thought (1918) translated the phrase “Fachwerk von Begriffen” as “framework of concepts, and Mayer and Tilman also use “framework” (2009) in their discussion of Hilbert’s Lectures on the Foundations of Physics. David Hilbert and the Axiomatization of Physics: From Grundlagen der Geometrie to Grundlagen der Physik, Springer, 2004. Also we should mention Corey’s use of “network of concepts” in his discussion (2004) of Hilbert’s axiomatization of Physics.
- 2.
Hilbert's Lectures on the Foundations of Physics, 1915–1927,bT. Sauer and U. Majer (eds.), Springer 2009, p. 420.
- 3.
D. Hilbert and P. Bernays, Grunndlagen Der Mathematik, vol 1, verlag von Julius Springer, 1934, pp. 5–7.
- 4.
The use of schematic letters rather than variables of the appropriate type, allows them to be replaced by specific predicates and relations. This would not be permitted if the places in the Fachwork were marked by variables of the appropriate type.
- 5.
Hilbert and Bernays, vol.1, p.6, Axiom II,1.
- 6.
This was a longstanding view of Hilbert. Recall that his list of open problems in 1900 included the sixth problem: Axiomatize all of Physics.
- 7.
Philosophical and Mathematical Corresspondence: Gottlob Frege, The university of Chicago Press, Chicago, 1980. Eds G. Ga briel, H. Hermes, F. Kambartel, c. Thiel, AS.Veraart. Abridged from the German edition by B. McGuinnness and translated by H. Kartel.
- 8.
7 Newman, M. H. A. (1928), “Mr. Russell’s ‘Causal Theory of Perception,’” Mind,37(146): 137–148.
- 9.
Chap. 7.
- 10.
“Ramsey's Dialetheism”, in The Law of Non-Contradiction, G. Priest, B. Armour-Garb, and J.C. Beall (eds.), Oxford University Press, 2005.
- 11.
A. Koslow, “The Representational Inadequacy of Ramsey Sentences.”, Theoria, vol LXX!II, 2006, Part 2.
- 12.
S. Morgenbesser and A. Koslow, “Theories and their Worth”, Journal of Philosophy 2012, pp.616–647.
- 13.
The uniformity involved in the various subsumtive applications is radically different from the deductive uniformity studied by P. Kitcher,”Explanatory Unification”, Philosophy of Science, 48, 507–31, and M. Friedman, “Explanation and Scientific Understanding”, Journal of Philosophy, 71(I): 5–19, 1974. For one thing the uniformity that Kitcher appeals to is a uniformity of a deductive pattern, while all the subsumed cases of a theory are not deductive consequences of that theory.
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Koslow, A. (2019). David Hilbert’s Architecture of Theories and Schematic Structuralism. In: Laws and Explanations; Theories and Modal Possibilities. Synthese Library, vol 410. Springer, Cham. https://doi.org/10.1007/978-3-030-18846-7_9
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