Philosophy Concordance - online philosophical quotations

During the first half of the present century a number of outstanding philosophers realized that language theory could profitably be viewed as far more than merely a means of studying one among the many human faculties, or merely sharpening the tool we use to philosophize - they realized that there is a sense in which philosophy of language comprises (almost) the whole of philosophy.

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.

A number is not one of the things it can be reduced to, it is rather what all of these things have in common, it 5 is a number .

Everyone of us can be confronted (at once, but also during the whole span of his life) with at most a finite number of objects.

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'.

This nature of infinity has been repeatedly pointed out - probably for the first time by Aristotle, and more recently by several of the most outstanding 9 mathematicians and philosophers of this century - but many theoreticians simply ignore it. However, if there are no real infinite sets beyond those grounded in a finite number of generating rules, then there are also no functions with infinite domains beyond functions defined via finite rules.

There is no way of defining a function on an infinite domain, save by giving its values for a finite number of basic elements and then giving a finite number of rules to compute values for new elements from those already computed.

In some simple artificial languages there can be truth without meaning; in a universal natural language comprising an infinite number of truths this is, however, not possible.

The terms '9' and 'the number of planets' name one and the same abstract entity but presumably must be regarded as unlike in meaning; for astronomical observation was needed, and not mere reflection on meanings, to determine the sameness of the entity in question.

The terms '9' and 'the number of the planets' name one and the same abstract entity but presumably must be regarded as unlike in meaning; for astronomical observation was needed, and not mere reflection on meanings, to determine the sameness of the entity in question.