[FOM] Brouwer on mathematical operations

Tero Tulenheimo tero.tulenheimo at helsinki.fi
Sun Jan 4 10:00:38 EST 2009

Speaking of "operational approaches" in the foundations of  
mathematics, perhaps the most obvious reference has not been  
mentioned, viz. Paul Lorenzen's "operative approach" to logic and  
mathematics developed in his monograph "Einführung in die operative  
Logik und Mathematik" (1955, second ed. 1969), Springer.

Lorenzen took mathematical theories to consist of applying mechanical  
operations to constellations of symbols in accordance with given  
rules. He did not develop the approach much beyond this book,  
especially not its parts concerned with logic; instead he went on to  
formulate the "dialogical foundation" of logic.

William Craig's and Thoralf Skolem's reviews of Lorenzen's book can be  
found via projecteuclid.org.

A recent article related to Lorenzen's operative approach is P.  
Schroeder-Heister's "Lorenzen's operative justification of  
intuitionistic logic," see


Kind regards,
   Tero Tulenheimo

> "One cannot inquire into the foundations and nature of mathematics
> without delving into the question of the operations by which the
> mathematical activity of the mind is conducted. If one failed to take
> that into account, then one would be left studying only the language
> in which mathematics is represented rather than the essence of
> mathematics."  L.E.J. Brouwer
> I am only a student, but I am inclined to agree with Brouwer.  Does
> anyone else agree also?  If I venture to call intuitionism an
> "operational" approach in foundations, can anyone think of anything
> similar existing today in foundations?  What operations do you think
> Brouwer had in mind?
> By the way, I would be much obliged to anyone who could tell me the
> source of this quotation.
> Sincerely,
> Lucius Schoenbaum

Tero Tulenheimo, DPhil
Researcher, University of Helsinki, Department of Philosophy
URL (official): http://www.helsinki.fi/filosofia/filo/henk/tulenheimo.htm
URL (personal): http://www.geocities.com/tero_tulenheimo

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