[FOM] Re. alleged quote by Hilbert (redirected from Michael Detlefsen)

[Vladimir Sazonov]

Michael Detlefsen <detlefsen.1 at nd.edu> 
presented some quotations from Hilbert, in particular containing: 

Hilbert:
> [Note: My [Michael Detlefsen] translation: "Mathematics is not like 
a game in which the 
> problems are determined by arbitrarily invented rules. Rather, it is a 
> conceptual system of inner necessity that can only be what it is and not 
> otherwise."]

Thus, not a game. 

"Conceptual system of inner necessity" seems to assume some system of 
formal reasoning revealing this "inner necessity". I do not see any 
other non-subjective mechanism (which should be coherent, of course, 
with some subjective intuition). 


Hilbert:
To make it a universal requirement 
> that each individual formula then be interpretable by itself is by no means 
> reasonable; on the contrary, a theory by its very nature is such that we do 
> not need to fall back upon intuition or meaning in the midst of some 
> argument. 

Thus, in some situations it is like a game with meaningless symbols... 

Even school children implicitly (and quite reasonably) use this approach 
when multiplying decimal numbers without thinking about the meaning of 
each intermediate formal step. Is not this an example of the universal 
idea on which the whole mathematics is based - automating the thought 
(even when we deduce - not just compute deterministically) by using 
formal rules? 


Hilbert:
> 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. 

Thus, a game? What should be the conclusion from all of this? Was 
Hilbert actually not a formalist? Or should we change the traditional 
(anekdotal - due to some opponents) opinion on what formalism is, really? 

(For me the above "contradictions" in Hilbert's notes are inessential. 
He had right philosophical idea which we should try to understand, 
at last.) 


Vladimir Sazonov



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