137
are known not to be instantiated by any object, such as lengths greater than the
diameter of the universe and masses greater than the mass of the universe).
36
Finally,
while a realist ontology has a greater categorical complexity, it spares the nominalist
requirement of an immensely great number of properties, immensely growing at
each time instant with the creation of new properties. For these reasons we maintain
here the position that individual properties are universals.
5.3.2 Individual properties as universals: an explanation
The idea that individual properties are universals is conceptually sophisticated: How
can it be that indistinguishable properties of distinct objects may correspond in fact
to the same individual property? Let us consider a mathematical relation such as ∑
1/(i 2
i
) = ln(2), where i is an integer ranging from 1 to infinity, an equation which is
known to be true, given that both ∑ 1/(i 2
i
) = 0.693147 … and ln(2) = 0.693147 ….
In terms of the involved numbers the relation ∑ 1/(i 2
i
) = ln(2) is not different from
0.693147 … = 0.693147 …: but while the latter is a logical identity, which does not
convey any information, the former implies some mathematical knowledge, so that
in some respect the two entities, ∑ 1/(i 2
i
) and ln(2), must be different. However,
there is also a respect in which the equality actually holds, so that we can say that
∑ 1/(i 2
i
) = ln(2) is true, while, for example, ∑ 1/(i 2
i
) = 2 is false. This double
interpretation—different but equal at the same time—accounts for the principled
difference between ∑ 1/(i 2
i
) = ln(2) and 0.693147 … = 0.693147 …, which can be
explained in terms of the distinction between the sense and the reference of an
expression (as noted in Box 5.1 above), where the sense of an expression is the concept it designates and the reference of an expression is the entity it refers to.
37
Hence “∑ 1/(i 2
i
)” and “ln(2)”
• are different expressions (the former starts with a symbol of summation, the latter with the name of a function, and so on),
• with different senses, since they designate different concepts (the former is a
series, the latter is a function evaluated in a given argument),
• but with the same referent, since they refer to the same mathematical object (the
number 0.693147 …).
This is a possible interpretation of the relation P[a i ] ≈ P[a j ], and indeed the one we
adopt here: when claiming, e.g., that the length of a i is indistinguishable from (or
36 There is one more reason supporting realism about properties, related to the status of values of
properties, a subject that we explore in Chap. 6. Just as a mention here, though obtained through
the conventional definition of a unit, an entity such as 1.2345 m seems to have an existence independent of the knowledge that we have of it. In other words, values are not concepts.
37 An analogous distinction is put between the intension and the extension of a concept (see Sect.
2.1). Hence, the intensions of the concepts <∑ 1/(i 2
i )> and are different, while their
extension is the same, i.e., the number 0.693147 …. Note that intensions and extensions are sometimes attributed to terms too; see, e.g., Chalmers, 2002.
5.3 A philosophical interlude
are known not to be instantiated by any object, such as lengths greater than the
diameter of the universe and masses greater than the mass of the universe).
36
Finally,
while a realist ontology has a greater categorical complexity, it spares the nominalist
requirement of an immensely great number of properties, immensely growing at
each time instant with the creation of new properties. For these reasons we maintain
here the position that individual properties are universals.
5.3.2 Individual properties as universals: an explanation
The idea that individual properties are universals is conceptually sophisticated: How
can it be that indistinguishable properties of distinct objects may correspond in fact
to the same individual property? Let us consider a mathematical relation such as ∑
1/(i 2
i
) = ln(2), where i is an integer ranging from 1 to infinity, an equation which is
known to be true, given that both ∑ 1/(i 2
i
) = 0.693147 … and ln(2) = 0.693147 ….
In terms of the involved numbers the relation ∑ 1/(i 2
i
) = ln(2) is not different from
0.693147 … = 0.693147 …: but while the latter is a logical identity, which does not
convey any information, the former implies some mathematical knowledge, so that
in some respect the two entities, ∑ 1/(i 2
i
) and ln(2), must be different. However,
there is also a respect in which the equality actually holds, so that we can say that
∑ 1/(i 2
i
) = ln(2) is true, while, for example, ∑ 1/(i 2
i
) = 2 is false. This double
interpretation—different but equal at the same time—accounts for the principled
difference between ∑ 1/(i 2
i
) = ln(2) and 0.693147 … = 0.693147 …, which can be
explained in terms of the distinction between the sense and the reference of an
expression (as noted in Box 5.1 above), where the sense of an expression is the concept it designates and the reference of an expression is the entity it refers to.
37
Hence “∑ 1/(i 2
i
)” and “ln(2)”
• are different expressions (the former starts with a symbol of summation, the latter with the name of a function, and so on),
• with different senses, since they designate different concepts (the former is a
series, the latter is a function evaluated in a given argument),
• but with the same referent, since they refer to the same mathematical object (the
number 0.693147 …).
This is a possible interpretation of the relation P[a i ] ≈ P[a j ], and indeed the one we
adopt here: when claiming, e.g., that the length of a i is indistinguishable from (or
36 There is one more reason supporting realism about properties, related to the status of values of
properties, a subject that we explore in Chap. 6. Just as a mention here, though obtained through
the conventional definition of a unit, an entity such as 1.2345 m seems to have an existence independent of the knowledge that we have of it. In other words, values are not concepts.
37 An analogous distinction is put between the intension and the extension of a concept (see Sect.
2.1). Hence, the intensions of the concepts <∑ 1/(i 2
i )> and
extension is the same, i.e., the number 0.693147 …. Note that intensions and extensions are sometimes attributed to terms too; see, e.g., Chalmers, 2002.
5.3 A philosophical interlude