Philosophy Concordance - online philosophical quotations

For the entities we mention to help specify a state of mind{ XE "mind" } do not have to play any psychological or epistemological role at all, just as numbers play no physical role.

Another prominent example is that of the meaning of numerals: all of Peano arithmetic does not allow us to distinguish whether a number is the set of equipotent sets (as Frege, in effect, maintained), or the set of all the numbers smaller than itself (as von Neumann proposed), or a primitive object.

There is little doubt that natural numbers do exist; at least to the extent to which objects which cannot be touched and tasted can exist.

And what can be said about numbers, can, more generally, be said about meanings: the meaning of an expression is not one or another definite thing conforming to our usage of the expressions within language; at most it may be said to be what all such things have in common.

We can never encounter the set of all natural numbers; we can at most encounter a rule of the kind of '0 is a number x is a number, then also x' is a number'.

We may define function f with its domain equal to the set of all natural numbers by defining f(0) and giving the recipe how to compute f(x') out of f(x), or we may base the definition on a more complicated system of rules generating natural numbers, but it is impossible to define such a function directly, bypassing generating rules altogether.

Following the usual praxis we quote Qittgenstein by paragraph numbers rather than page numbers.

As Quine puts it: "there is no saying absolutely what the numbers are; there is only arithmetic" (Quine 1969: 45).

Carnap's suggestions on this subject were, despite their sketchiness, very suggestive. He explained spatio-temporal point-instants as quadruples of real numbers and envisaged assignment of sense qualities to point-instants according to certain canons.

Imagine, for the sake of analogy, that we are given the rational numbers.