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Full Idea
The statement that a class is to be represented by a symbol is a definition in mathematics, and says nothing about mathematical entities. Any formula can be stated in terms of primitive ideas, so the definitions are superfluous.
Gist of Idea
In mathematics definitions are superfluous, as they name classes, and it all reduces to primitives
Source
Bertrand Russell (The Principles of Mathematics [1903], §412)
Book Ref
Russell,Bertrand: 'Principles of Mathematics' [Routledge 1992], p.429
A Reaction
[compressed wording] I'm not sure that everyone would agree with this (e.g. Kit Fine), as certain types of numbers seem to be introduced by stipulative definitions.
| 602 | Some fools think you cannot define anything, but only say what it is like [Antisthenes (I), by Aristotle] |
| 5619 | No a priori concept can be defined [Kant] |
| 22201 | The use of mathematical-style definitions in philosophy is fruitless and harmful [Husserl] |
| 14115 | Definition by analysis into constituents is useless, because it neglects the whole [Russell] |
| 14159 | In mathematics definitions are superfluous, as they name classes, and it all reduces to primitives [Russell] |
| 12619 | We have no successful definitions, because they all use indefinable words [Fodor] |
| 9331 | How do we determine which of the sentences containing a term comprise its definition? [Horwich] |
| 12585 | Most people can't even define a chair [Peacocke] |
| 14981 | Philosophical concepts are rarely defined, and are not understood by means of definitions [Sider] |
| 15015 | It seems possible for a correct definition to be factually incorrect, as in defining 'contact' [Sider] |
| 20391 | Feminists warn that ideologies use timeless objective definitions as a tool of repression [Davies,S] |