NDLI: What is the axiomatic method?

Content Provider	Springer Nature Link
Author	Hintikka, Jaakko
Copyright Year	2009
Abstract	The modern notion of the axiomatic method developed as a part of the conceptualization of mathematics starting in the nineteenth century. The basic idea of the method is the capture of a class of structures as the models of an axiomatic system. The mathematical study of such classes of structures is not exhausted by the derivation of theorems from the axioms but includes normally the metatheory of the axiom system. This conception of axiomatization satisfies the crucial requirement that the derivation of theorems from axioms does not produce new information in the usual sense of the term called depth information. It can produce new information in a different sense of information called surface information. It is argued in this paper that the derivation should be based on a model-theoretical relation of logical consequence rather than derivability by means of mechanical (recursive) rules. Likewise completeness must be understood by reference to a model-theoretical consequence relation. A correctly understood notion of axiomatization does not apply to purely logical theories. In the latter the only relevant kind of axiomatization amounts to recursive enumeration of logical truths. First-order “axiomatic” set theories are not genuine axiomatizations. The main reason is that their models are structures of particulars, not of sets. Axiomatization cannot usually be motivated epistemologically, but it is related to the idea of explanation.
Starting Page	69
Ending Page	85
Page Count	17
File Format	PDF
ISSN	00397857
Journal	Synthese
Volume Number	183
Issue Number	1
e-ISSN	15730964
Language	English
Publisher	Springer Netherlands
Publisher Date	2009-10-09
Publisher Place	Dordrecht
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Axiomatic method Information Logical consequence Completeness Set theory Philosophy of Science Epistemology Philosophy of Language Logic Metaphysics
Content Type	Text
Resource Type	Article
Subject	Philosophy Social Sciences

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