Hilbert's program

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        <b>Hilbert's program</b>
      
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        Hilbert's program was based upon the construction of formal processes out of material 
        tokens
      
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        ("extralogical concrete objects") and physical operations on them. While he believed 
        infinite collections of
      
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        such tokens were not physically realizable, he did believe that consistent finite 
        representations of infinite
      
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        entities could be constructed and implemented through finite sets of physical tokens.
      
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        As a further precondition for using logical deduction and carrying out logical operations, 
        something must be
      
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        given in conception, viz., certain extralogical concrete objects which are intuited as 
        directly experienced prior to all
      
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        thinking. For logical deduction to be certain, we must be able to see every aspect of 
        these objects, and their properties,
      
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        differences, sequences, and contiguities must be given, together with the objects 
        themselves, as something which
      
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        cannot be reduced to something else and which requires no reduction. This is the basic 
        philosophy which I find
      
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        necessary, not just for mathematics, but for all scientific thinking, understanding, and 
        communicating. The subject
      
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        matter of mathematics is, in accordance with this theory, the concrete symbols themselves 
        whose structure is
      
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        immediately clear and recognizable. (Hilbert, 1925, p.192)
      
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        These concrete tokens or primitives would then be combined in various ways to construct 
        more
      
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        complicated relations between the combinations. Von Neumann, who was associated with 
        Hilbert's
      
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        formalist program neatly summarized the goals of the movement:
      
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        The leading idea of Hilbert's theory of proof is that, even if the statements of classical 
        mathematics should turn out
      
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        to be false as to content, nevertheless, classical mathematics involves an internally 
        closed procedure which operates
      
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        according to fixed rules known to all mathematicians and which consists basically in 
        constructing successively certain
      
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        combinations of primitive symbols which are considered "correct" or "proved." ...We must 
        regard classical
      
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        mathematics as a combinatorial game played with the primitive symbols, and we must 
        determine in a finitary
      
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        combinatorial way to which combinations of primitive symbols the construction methods or 
        "proofs" lead." (von
      
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        Neumann, 1931)
      
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        The goal of formalist mathematics was to construct a system of operations on tokens which 
        was internally
      
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        consistent, that is, no combination of allowed operations should lead to a contradiction.
      
      
        
      
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