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Single Idea 18092

[filed under theme 6. Mathematics / A. Nature of Mathematics / 5. The Infinite / l. Limits ]

Full Idea

After Weierstrass had stressed the importance of limits, one now needed to be able to prove the existence of such limits.

Gist of Idea

Weierstrass made limits central, but the existence of limits still needed to be proved

Source

report of Karl Weierstrass (works [1855]) by David Bostock - Philosophy of Mathematics 4.4

Book Ref

Bostock,David: 'Philosophy of Mathematics: An Introduction' [Wiley-Blackwell 2009], p.98

A Reaction

The solution to this is found in work on series (going back to Cauchy), and on Dedekind's cuts.

The 6 ideas with the same theme [the conclusion of a converging series]:

18082 Quantities and ratios which continually converge will eventually become equal [Newton]
18084 When successive variable values approach a fixed value, that is its 'limit' [Cauchy]
18092 Weierstrass made limits central, but the existence of limits still needed to be proved [Weierstrass, by Bostock]
18087 If x changes by less and less, it must approach a limit [Dedekind]
18187 Theorems about limits could only be proved once the real numbers were understood [Maddy]
22886 The modern idea of 'limit' allows infinite quantities to have a finite sum [Bardon]