[FOM] Brouwer on mathematical operations
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
> "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.
> 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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