Reflections on Hilbert’s Program

  • Wilfried Sieg
Part of the Synthese Library book series (SYLI, volume 211)


Hilbert’s Program deals with the foundations of mathematics from a very special perspective; a perspective that stems from Hubert’s answer to the question “What Is Mathematics?”. The popular version of his “formalist” answer, radical at Hubert’s time and shocking to thoughtful mathematicians even today, is roughly this: the whole “thought-content” of mathematics can be uniformly expressed in a comprehensive formal theory, mathematical activity reduces to the manipulation of symbolic expressions, and mathematics itself is just “ein Formelspiel”. Hubert defended his “playful” view of mathematics against intuitionistic attack by remarking:

The formula game that Brouwer so deprecates has, besides its mathematical value, an important general philosophical significance. For this formula game is carried out according to certain definite rules, in which the Technique of Our Thinking is expressed. These rules form a closed system that can be discovered and definitively stated. The fundamental idea of my proof theory is none other than to describe the activity of our understanding, to make a protocol of the rules according to which our thinking actually proceeds. Thinking, it so happens, parallels speaking and writing: we form statements and place them one behind the another. If any totality of observations and phenomena deserves to be made the object of serious and thorough investigation, it is this one—...1


Proof Theory Intuitionistic Theory Generate Clause Consistency Proof Elementary Number Theory 
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  1. 1.
    Hubert , “Die Grundlagen der Mathematik” (1927) This paper was presented to the Mathematical Seminar in Hamburg in July 1927; it was published in: Abhandlungen aus dem mathematischen Seminar der Hamburgischen Universität 6 (1928), pp. 65–85. A translation can be found in van Heijenoort (ed.), From Frege to Gödel, Cambridge, 1967, 464–479.—Incidentally, it was Hubert who provoked— by the very formulation of the Entscheidungsproblem for predicate logic—Church’s and Turing’s work on computability, work that is central in theoretical computer science.Google Scholar
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    Bernays, “Die schematische Korrespondenz und die idealisierten Strukturen”, reprinted in the volume of essays mentioned in15, p. 186. This paper was published originally in 1970. I should mention that essentially the same view is expressed in Bernays’s 1930 paper mentioned above; except, that the finitist standpoint is taken as “absolute”. After the discussion of completeness (“deduktive Abgeschlossenheit”) inclusive a footnote stating (incorrectly, as we know) that completeness of a formal theory is not as far reaching as its decidability, we read (on p. 59): Im Gebiete dieser und verwandter Fragen liegt noch ein beträchtliches Feld der Problematik offen. Diese Problematik ist aber nicht von der Art, daß sie eine Einwendung gegen den von uns eingenommenen Standpunkt darstellt. Wir müssen uns nur gegenwärtig halten, daß der Formalismus der Sätze und Beweise, mit denen wir unsere Ideenbildung zur Darstellung bringen, nicht zusammenfällt mit dem Formalismus derjenigen Struktur, die wir in der Gedankenbildung intendieren. Der Formalismus reicht aus, um unsere Ideen von unendlichen Mannigfaltigkeiten zu formulieren, aber er vermag im allgemeinen nicht, die Mannigfaltigkeit gleichsam aus sich kombinatorisch zu erzeugen.Google Scholar
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    There is a genuine conceptual problem for attempts to extend the reductive work concerning impredicative subsystems of analysis beyond the strongest system that has been treated: i.d. classes are provably not sufficient. We have to find a new, broader concept of “constructive mathematical object”, if there is to be any prospect for genuine advances.Google Scholar
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    One has to keep in mind that for Hubert there was only one constructive mathematics, as finitist and intuitionist mathematics were assumed to be coextensive.Google Scholar

Copyright information

© Kluwer Academic Publishers 1990

Authors and Affiliations

  • Wilfried Sieg
    • 1
  1. 1.Department of PhilosophyCarnegie Mellon UniversityPittsburghUSA

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