more from this thinker | more from this text
Full Idea
It used to be common to define numbers by means of 1, with 2 being 1+1 and so on. But this method was only applicable to finite numbers, made a tiresome different between 1 and the other numbers, and left + unexplained.
Gist of Idea
Numbers were once defined on the basis of 1, but neglected infinities and +
Source
Bertrand Russell (The Principles of Mathematics [1903], §109)
Book Ref
Russell,Bertrand: 'Principles of Mathematics' [Routledge 1992], p.112
A Reaction
Am I alone in hankering after the old approach? The idea of a 'unit' is what connected numbers to the patterns of the world. Russell's approach invites unneeded platonism. + is just 'and', and infinities are fictional extrapolations. Sounds fine to me.
| 11041 | Some quantities are discrete, like number, and others continuous, like lines, time and space [Aristotle] |
| 17843 | The idea of 'one' is the foundation of number [Aristotle] |
| 17850 | Each many is just ones, and is measured by the one [Aristotle] |
| 17851 | Number is plurality measured by unity [Aristotle] |
| 1600 | Euclid's common notions or axioms are what we must have if we are to learn anything at all [Euclid, by Roochnik] |
| 17783 | A number is not a multitude, but a unified ratio between quantities [Newton] |
| 9800 | Arithmetic is based on definitions, and Sums of equals are equal, and Differences of equals are equal [Mill] |
| 14116 | Numbers were once defined on the basis of 1, but neglected infinities and + [Russell] |
| 10205 | Mathematics originally concerned the continuous (geometry) and the discrete (arithmetic) [Shapiro] |
| 19093 | Greek mathematics is wholly sensory, where ours is wholly inferential [Macbeth] |