Artículo que explica los fundamentos de la formalización matemática: por qué es necesaria, y por qué es mucho más difícil que las explicaciones intuitivas.
The Narrow Road » Blog Archive » On Abstraction The point of this rigmarole is to show that ‘1 + 1 = 2′
can be enunciated without mention of either ‘1′ or
‘2′. The point may become clearer if we take an
illustration. Suppose Mr A has one son and one daughter. It is required
to prove that he has two children. We intend to state the premise and
the conclusion in a way not involving the words ‘one’ or
‘two’.
We translate the above general statement by putting:
φx. = .x is a son of Mr A,
ψx. = .x is a daughter of Mr A.
Then there is an object having the property φ, namely Mr A junior; whatever x
may be, if it has some property that Mr A junior does not have, it is
not Mr A junior, and therefore not a son of Mr A senior. This is what
we mean by saying that ‘being a son of Mr A’ is a unit
property. Similarly ‘being a daughter of Mr A’ is a unit
property. Now consider the property ‘being a son or daughter of
Mr A’, which we will call χ. There are objects, the son and
daughter, of which (1) the son has the property of being male, which
the daughter has not; (2) the son has the property χ and the
daughter has the property χ; (3) if x is an object which lacks some property possessed by the son and also some property possessed by the daughter, then x
is not a son or a daughter of Mr A. It follows that χ is a dual
property. In short … a man who has one son and one daughter has
two children.
