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I am wondering if Dedekind's theory about the structure of deductive science influenced the work of Hilbert.

Hilbert obviously favored axioms at the beginnings of a deductive science, whereas Dedekind seemed to take a more conceptual approach.

Any details about the relationship between the works of these mathematicians would be appreciated.

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Hilbert was influenced especially by Dedekind's 1888 essay The Nature and Meaning of Numbers, but he shared only half of Dedekind's approach. To keep things in perspective, it is important to note the third, most traditional genetic approach, advocated at the time by Helmholtz-Kronecker and more rigorously by Frege, as reflected in the Frege-Hilbert correspondence. Hilbert explicitly distinguished his own axiomatic approach from Dedekind's structural one, which, arguably, combined axiomatic and genetic one. The combination came to be appreciated much later, when Gödel showed that axiomatic theories could not be "grounded" the way Hilbert hoped.

Dedekind and Hilbert were both promoting abstract arguments about objects with multiple concrete realizations, on this they were on the same side. But Dedekind's approach was bottom-up, abstracting general properties from concrete realizations and operating on them, while Hilbert's was top-down, laying down the axioms and deriving theorems about 'incomplete' objects. He saw the genetic side as redundant, at least, from the foundational perspective. Modernizing, we could say that Dedekind had in mind a class of models, while Hilbert had a formal theory. In a similar spirit, some modern mathematicians distinguish between "structural" axioms of "working mathematicians" and "foundational" axioms of logicians, e.g. Feferman in Does mathematics need new axioms?

Interconnections between Hilbert's and Dedekind's ideas are analyzed in Sieg-Schlimm, Dedekind's Analysis of Number: Systems and Axioms:

"In 1888 Hilbert made his Rundreise from Konigsberg to other German university towns. He arrived in Berlin just as Dedekind's Was sind und was sol/en die Zahlen? had been published. Hilbert reports that in mathematical circles everyone, young and old, talked about Dedekind's essay, but mostly in an opposing or even hostile sense. A year earlier, Helmholtz and Kronecker had published articles on the concept of number in a Festschrift for Eduard Zeller... Dedekind's reflections... can be understood... as articulating an axiomatic approach that is joined with a genetic one in a methodologically coherent way... If we think of the genetic method as underlying the construction of mathematical objects, systems of which are models of appropriate axiom systems, we can see very clearly how it complements in Dedekind's hands an axiomatic approach.

In his essay Uber den Zahlbegriff, Hilbert distinguished sharply between the axiomatic and genetic method, but did not recognize then the complementary roles they play for the foundations of arithmetic... Dedekind's approach is associated with a novel structuralist perspective on mathematics and is grounded in logic broadly conceived. Hilbert sustains this general perspective in what he later calls existential axiomatics, but he gives up a logicist in favor of a finitist grounding of mathematics... By tracing its development we provide a view of Dedekind's evolving foundational position that apparently differs from Hilbert's: to our knowledge, Hilbert never considered Dedekind as having used the axiomatic method.

[...] There is no conflict, and consequently no choice has to be made, between a genetic and an axiomatic approach for Dedekind... There is no supersession of Dedekind's "deductive method" (described on pp. 246-248) by the axiomatic method of Hilbert's, but the former is rather the very root of the latter. Hilbert's first axiomatic formulations in Uber den Zahlbegriff and Grundlagen der Geometrie are patterned after Dedekind's. Indeed, Hilbert is a logicist in Dedekind's spirit at that point, and it is no accident that, as late as 1917/1918, he was attracted by attempts to provide a logicist foundation of mathematics."

As Sieg-Schlimm admit, their interpretation is controversial and other historians, Ferreiros, Corry and McCarty see them as further apart, and Dedekind as more "non-modern", "non-axiomatic" and "Kantian". However, Hilbert himself retained Kantian elements in his treatment of symbol manipulation, see Was there a Kantian influence on Hilbert's formalist programme?, and Blanchette, writing for SEP, also cites their commonalities:

"Perhaps most clearly illustrated in Dedekind 1888, the central idea of the new approach is to understand mathematical theories as characterizing general “structural” conditions that might be had in common by any number of different ordered domains. Just as, in algebra, the axioms for a group give general conditions that can be satisfied by any manner of object whatsoever under appropriate relations, so too on the new view the axioms of geometry specify multiply-instantiable conditions. Viewing theories from this modern perspective, it is entirely appropriate to take axioms as Hilbert does, since reinterpretable sentences are the right vehicles to express the multiply-instantiable conditions in question. From the point of view of the earlier fixed-domain conception of theories, on the other hand, such reinterpretable sentences are entirely inappropriate as axioms, since they fail to fix a determinate subject-matter."

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  • $\begingroup$ Amazing answer... Thanks so much for providing this detail. Do you mind giving further insight on the "top down" and "bottom up" concepts you mention? $\endgroup$
    – Demon
    Mar 1 at 10:06
  • $\begingroup$ @Demon Those are just informal descriptions. Dedekind tended to start from simple structures, like integers, and construct more complex fields and rings from them, then codify common properties into definitions and axioms of general theories (of ideals in rings, etc.). These constructions ensured that general theories are sensible since their building blocks are known to be. Hilbert wanted sensibility (consistency) of axiomatic theories ensured independently of their genesis, by finite means based on their formal symbolic representation. $\endgroup$
    – Conifold
    Mar 1 at 10:20

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