Crítica, Revista Hispanoamericana de Filosofía, Volumen 25, número 73, abril 1993
Sistemas de cálculo como formas de
Logicismo
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Ángel Nepomuceno Fernández
Departamento de Filosofía y Lógica y Filosofía de la Ciencia
Universidad de Sevilla
Resumen: The logicism may be regarded like a fossil stone that has not
utility nowadays. In this sense, logicism took care of the research
about the foundations of mathematics but apparently its task
arrived at its end many years ago because of sorne results that
were eetablished during the century. However it is not wholly
right. Understanding logicism as an attempt to reduce classical
mathematics to logic means we can distinguish: 1) the idea
according to which mathematic is logic in sorne way, and 2) a
metaphysical program of research to: a) define mathematical
notions as logical notions, and b) show that the mathematical
theorems are logical theorems.
The failure (if so) concerned to 2), since 1) was assumed
by many logicians. Recovering logicism is not easy and there
may be several ways. One of them is the one followed by N.B.
Cocchiarella whose systems (there are more than one) represent a
form of logicism (Frege's or Russell's form). From those systems
though a bit changed from my own point of view we can
define a modal calcule that may have application in computer
science, what would not be a stale work.
From a common language we take in account two systems in
order to show that Cocchiarella's modified system is as powerful
deductively as that of Church modified functional second order
calcule. We can obtain new systems that represent form of
logicism and are more powerful than that of Church enlarging
Cochiarella's modified system. These new systems, that becomes
modal systems provided that one adds appropiate modal tools
(then they may be used in computer science), may be useful
to study logicism itself (as historical philosophy of logic and
mathematics).
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