Difference between Hilbert's program and Russell & Whitehead's Principia Mathematica Can someone explain to me the difference between Hilbert's program and Russell & Whitehead's Principia Mathematica? I know both of them wanted to reduce the mathematics into a set of axioms and inference rules but eventually Goedel dashed their hope completely by presenting his first and second incompleteness theorem. I need to know what is the difference between those two projects?
Thanks a lot.
 A: In spite of what is usually "divulgated", Hilbert's Program dates back to the beginning of 20th Century :

Although Hilbert proposed his program in this form only in 1921, various facets of it are rooted in foundational work of his going back until around 1900, when he first pointed out the necessity of giving a direct consistency proof of analysis.

After his successfull axiomatization of Euclidean geometry (1899), in his well-known list of twenty-three problems in mathematics of 1900 : “Mathematische Probleme”, Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Math.-Phys.Klasse, 253–297 (lecture given at the International Congress of Mathematicians in Paris, 1900), asked for :

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*2nd problem : Prove that the axioms of arithmetic are consistent


*6th problem : Mathematical treatment of the axioms of physics.
We can see also :

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*David Hilbert, “Über die Grundlagen der Logik und der Arithmetik”, in Verhandlungen des dritten Internationalen Mathematiker-Kongresses in Heidelberg, August 1904; English translation in Jean van Heijenoort, From Frege to Gödel : A Source Book in Mathematical Logic (1967), pages 129–138.


Work on the program progressed significantly in the 1920s with contributions from logicians such as Paul Bernays, Wilhelm Ackermann, John von Neumann, and Jacques Herbrand. It was also a great influence on Kurt Gödel, whose work on the incompleteness theorems were motivated by Hilbert's Program.

The work of Whitehead & Russell started after the pubblication of Russell's The Principles of Mathematics (1903) and culminated in the "gigantic" effort of the Principia Mathematica : 3 vols, 1910-1913.
Of course, the works of the members of the Hilbert's school during the 20's was highly influenced by the Principia; see :

*

*Paolo Mancosu, The Adventure of Reason : Interplay Between Philosophy of Mathematics and Mathematical Logic, 1900-1940 (2010), Part II : Foundations of Mathematics, page 121-on.


In brief, Principia was developed in the context of the philosophical view called Logicism :

the view that (some or all of) mathematics can be reduced to (formal) logic. It is often explained as a two-part thesis. First, it consists of the claim that all mathematical truths can be translated into logical truths or, in other words, that the vocabulary of mathematics constitutes a proper subset of the vocabulary of logic. Second, it consists of the claim that all mathematical proofs can be recast as logical proofs or, in other words, that the theorems of mathematics constitute a proper subset of the theorems of logic. As Russell writes, it is the logicist's goal

“to show that all pure mathematics follows from purely logical premises and uses only concepts definable in logical terms” (My Philosophical Development,1959, page 74).


The consistency of the system developed by W&R must be ensured by the choice of a limited set of postulates of "pure logic" that must be certain.
But the "technical" foundational work asked for the adoption of solutions, like the ramified type theory, and some axioms, like the Multiplicative Axiom (a form of Axiom of Choioce) and the Axiom of reducibility which, in spite of their "plausibility", were far from being certain on the basis of pure logic.

The role of logic in Hilbert's foundational program was very different.

The publication of Russell and Whitehead's Principia Mathematica provided the required logical basis for a renewed attack on foundational issues. [...] In September 1917, [hilbert] delivered an address to the Swiss Mathematical Society entitled “Axiomatic Thought” (1918). It is his first published contribution to mathematical foundations since 1905. In it, he again emphasizes the requirement of consistency proofs for axiomatic systems:

“The chief requirement of the theory of axioms must go farther [than merely avoiding known paradoxes], namely, to show that within every field of knowledge contradictions based on the underlying axiom-system are absolutely impossible.”

He poses the proof of the consistency of arithmetic (and of set theory) again as the main open problems. In both these cases, there seems to be nothing more fundamental available to which the consistency could be reduced other than logic itself. And Hilbert then thought that the problem had essentially been solved by Russell's work in Principia. Nevertheless, other fundamental problems of axiomatics remained unsolved, including the problem of the “decidability of every mathematical question,” which also traces back to Hilbert's 1900 address.

[...]

Within the next few years, however, Hilbert came to reject Russell's logicist solution to the consistency problem for arithmetic. [... In 1922] Hilbert presented his own proposal for a solution to the problem of the foundation of mathematics. This proposal incorporated Hilbert's ideas from 1904 regarding direct consistency proofs, his conception of axiomatic systems, and also the technical developments in the axiomatization of mathematics in the work of Russell as well as the further developments carried out by him and his collaborators. What was new was the way in which Hilbert wanted to imbue his consistency project with the philosophical significance necessary to answer [intuitionistic] criticisms: the finitary point of view.
According to Hilbert, there is a privileged part of mathematics, contentual elementary number theory, which relies only on a “purely intuitive basis of concrete signs.”

Simplifying a lot, the "ultimate" foundation of mathematics is grounded on the "contentual" elementary part of arithmetic.
